Communications method and apparatus

ABSTRACT

Communications method and apparatus include encoding information into a high-peakedness designed pulse train, converting the designed pulse train into a low-peakedness signal suitable for modulating a narrowband carrier to generate a physical communication signal with desired spectral and temporal properties, and generating and transmitting the physical communication signal. The communications method and apparatus also include receiving and demodulating the physical communication signal, and further converting the demodulated signal into a high-peakedness received pulse train corresponding to the designed pulse train, so that the encoded information may be extracted from the received pulse train.

CROSS REFERENCES TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. Pat. Application17/345,069, filed on 11 Jun. 2021, which is a continuation-in-part ofthe U.S. Pat. Application 16/858,603, filed on 25 Apr. 2020 (now U.S.Pat. 11,050,591), which is a continuation-in-part of the U.S. Pat.Application 16/383,782, filed on 15 Apr. 2019 (now U.S. Pat.10,637,490), which is a continuation-in-part of the U.S. Pat.Application 15/865,569, filed on 9 Jan. 2018 (now U.S. Pat. 10,263,635).This application is also related to the U.S. Provisional Pat.Applications 62/444,828 (filed on 11 Jan. 2017) and 62/569,807 (filed on9 Oct. 2017).

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

None.

COPYRIGHT NOTIFICATION

Portions of this patent application contain materials that are subjectto copyright protection. The copyright owner has no objection to thefacsimile reproduction by anyone of the patent document or the patentdisclosure, as it appears in the Patent and Trademark Office patent fileor records, but otherwise reserves all copyright rights whatsoever.

TECHNICAL FIELD

The present invention relates in general to the field of datacommunications and, in particular, to methods and correspondingapparatus for communicating over longer distances at lower power andenergy dissipation. This invention further relates to nonlinear signalprocessing, and, in particular, to method and apparatus for mitigationof outlier noise in the process of analog-to-digital conversion.

BACKGROUND

An outlier is something “abnormal” that “sticks out”. For example, thenoise that “protrudes” from background noise. Such noise would typicallybe, in terms of its amplitude distribution, non-Gaussian. What isactually observed may depend on a source, the way noise couples into asystem, and where in the system it is observed. Hence various particularinstances of outlier noise may be known under different names,including, but not limited to, such as impulsive noise, transient noise,sparse noise, platform noise, burst noise, crackling noise, clicks &pops, and others. Depending on the way noise couples into a system andwhere in the system it is observed, noise with the same origin may havedifferent appearances, and may or may not even be seen as an outliernoise.

Non-Gaussian (and, in particular, outlier) noise affecting communicationand data acquisition systems may originate from a multitude of naturaland technogenic (man-made) phenomena in a variety of applications.Examples of natural outlier (e.g. impulsive) noise sources include icecracking (in polar regions) and snapping shrimp (in warmer waters)affecting underwater acoustics [1-3]. Electrical man-made noise istransmitted into a system through the galvanic (direct electricalcontact), electrostatic coupling, electromagnetic induction, or RFwaves. Examples of systems and services harmfully affected bytechnogenic noise include various sensor, communication, and navigationdevices and services [4-15], wireless internet [16], coherent imagingsystems such as synthetic aperture radar [17], cable, DSL, and powerline communications [18-24], wireless sensor networks [25], and manyothers. An impulsive noise problem also arises when devices based on theultra-wideband (UWB) technology interfere with narrowband communicationsystems such as WLAN [26] or CDMA-based cellular systems [27]. Aparticular example of non-Gaussian interference is electromagneticinterference (EMI), which is a widely recognized cause of receptionproblems in communications and navigation devices. The detrimentaleffects of EMI are broadly acknowledged in the industry and includereduced signal quality to the point of reception failure, increased biterrors which degrade the system and result in lower data rates anddecreased reach, and the need to increase power output of thetransmitter, which increases its interference with nearby receivers andreduces the battery life of a device.

A major and rapidly growing source of EMI in communication andnavigation receivers is other transmitters that are relatively close infrequency and/or distance to the receivers. Multiple transmitters andreceivers are increasingly combined in single devices, which producesmutual interference. A typical example is a smartphone equipped withcellular, WiFi, Bluetooth, and GPS receivers, or a mobile WiFi hotspotcontaining an HSDPA and/or LTE receiver and a WiFi transmitter operatingconcurrently in close physical proximity. Other typical sources ofstrong EMI are on-board digital circuits, clocks, buses, and switchingpower supplies. This physical proximity, combined with a wide range ofpossible transmit and receive powers, creates a variety of challenginginterference scenarios. Existing empirical evidence [8, 28, 29] and itstheoretical support [6, 7, 10] show that such interference oftenmanifests itself as impulsive noise, which in some instances maydominate over the thermal noise [5, 8, 28].

A simplified explanation of non-Gaussian (and often impulsive) nature ofa technogenic noise produced by digital electronics and communicationsystems may be as follows. An idealized discrete-level (digital) signalmay be viewed as a linear combination of Heaviside unit step functions[30]. Since the derivative of the Heaviside unit step function is theDirac δ-function [31], the derivative of an idealized digital signal isa linear combination of Dirac δ-functions, which is a limitlesslyimpulsive signal with zero interquartile range and infinite peakedness.The derivative of a “real” (i.e. no longer idealized) digital signal maythus be viewed as a convolution of a linear combination of Diracδ-functions with a continuous kernel. If the kernel is sufficientlynarrow (for example, the bandwidth is sufficiently large), the resultingsignal would appear as an impulse train protruding from a continuousbackground signal. Thus impulsive interference occurs “naturally” indigital electronics as “di/dt” (inductive) noise or as the result ofcoupling (for example, capacitive) between various circuit componentsand traces, leading to the so-called “platform noise” [28]. Additionalillustrative mechanisms of impulsive interference in digitalcommunication systems may be found in [6-8, 10, 32].

The non-Gaussian noise described above affects the input (analog)signal. The current state-of-art approach to its mitigation is toconvert the analog signal to digital, then apply digital nonlinearfilters to remove this noise. There are two main problems with thisapproach. First, in the process of analog-to-digital conversion thesignal bandwidth is reduced (and/or the ADC is saturated), and aninitially impulsive broadband noise would appear less impulsive [7-10,32]. Thus its removal by digital filters may be much harder to achieve.While this may be partially overcome by increasing the ADC resolutionand the sampling rate (and thus the acquisition bandwidth) beforeapplying digital nonlinear filtering, this further exacerbates thememory and the DSP intensity of numerical algorithms, making themunsuitable for real-time implementation and treatment of non-stationarynoise. Thus, second, digital nonlinear filters may not be able to workin real time, as they are typically much more computationally intensivethan linear filters. A better approach would be to filter impulsivenoise from the analog input signal before the analog-to-digitalconverter (ADC), but such methodology is not widely known, even thoughthe concepts of rank filtering of continuous signals are well understood[32-37].

Further, common limitations of nonlinear filters in comparison withlinear filtering are that (1) nonlinear filters typically have variousdetrimental effects (e.g., instabilities and intermodulationdistortions), and (2) linear filters are generally better than nonlinearin mitigating broadband Gaussian (e.g. thermal) noise.

As the use and necessity of communications grows, the development ofsecure communications has become a priority to enable the use of various(e.g. wireless or wired) communication links without fear ofcompromising secure information. As cryptography is the standard way ofensuring security of a communication channel, steganography steps in toprovide even stronger assumptions. Thus, in the case of cryptology, anattacker cannot obtain information about the payload while inspectingits encrypted content. In the case of steganography, one cannot provethe existence of the covert communication itself. The purpose ofsteganography is to hide the very presence of communication by embeddingmessages into innocuous-looking cover objects, such as digital images.To accommodate a secret message, the original message, also called thecover message, or cover signal, is slightly modified by the embeddingalgorithm to obtain the stego signal. In steganography, the cover signalis a mere decoy and has no relationship to the hidden data.

The most important requirement for a steganographic system isundetectability: stego signals should be statistically indistinguishablefrom cover signals. In other words, there should be no artifacts in thestego signal that could be detected by an attacker with probabilitybetter than random guessing, given the full knowledge of the way theembedding of the hidden data is performed, including the statisticalproperties of the source of cover signals, except for the stego key.

While in steganography the information is hidden or embedded into acover signal, a covert channel allows parties to communicate “unseen,”hiding the very fact that communication is even occurring.

The additive white Gaussian noise (AWGN) capacity C of a channeloperating in the power-limited regime (i.e. when the receivedsignal-to-noise ratio (SNR) is small, SNR << 0 dB) may be expressed as C≈ P/(N₀ ln 2), where P is the average received power and N₀ is the powerspectral density (PSD) of the noise. This capacity is linear in powerand insensitive to bandwidth and, therefore, by spreading the averagetransmitted power of the information-carrying signal over a largefrequency band, the average PSD of the signal could be made much smallerthan the PSD of the noise. This would “hide” the signal in the channelnoise, making the transmission covert and insensitive to narrowbandinterference.

One of the common ways to achieve such “spreading” is frequency-hoppingspread spectrum (FHSS) [38]. This technique is widely used, for example,in legacy military equipment for low-probability-of-intercept (LPI)communications. However, using frequency hopping for covertcommunications is nearly obsolete today, since modern widebandsoftware-defined radio (SDR) receivers may capture all of the hops andput them back together.

Another common and widely used spread-spectrum modulation technique isdirect-sequence spread spectrum (DSSS) [39]. In DSSS, the narrow-bandinformation-carrying signal of a given power is modulated by awider-band, unit-power pseudorandom signal known as a spreadingsequence. This results in a signal with the same total power but alarger bandwidth, and thus a smaller PSD. After demodulation(“de-spreading”) in the receiver, the original information-carryingsignal is restored. However, such demodulation requires a precisesynchronization, which is perhaps the most difficult and expensiveaspect of a DSSS receiver design. Also, while de-spreading may not beperformed without the knowledge of the spreading sequence by thereceiver, the spreading code by itself may not be usable to secure thechannel. For example, linear spreading codes are easily decipherableonce a short sequential set of chips from the sequence is known. Toimprove security, it would be desirable to perform a “code hopping” in amanner akin to the frequency hopping. However, synchronization may be anextremely slow process for pseudorandom sequences, especially for largespreading waveforms (long codes), and thus such DSSS code hopping may bedifficult to realize in practice.

In the power-limited regime, we would normally use binary coding andmodulation (e.g. binary phase-shift keying (BPSK) or quadraturephase-shift keying (QPSK)) for the narrow-band information-carryingsignal, and this signal would be significantly oversampled to enablewideband spreading. Thus an idealized narrow-band information-carryingsignal that is to be “spread” may be viewed as a discrete-level signalthat is a linear combination of analog Heaviside unit step functions[30] delayed by multiples of the bit duration. Such a signal would havea limited bandwidth and a finite power. Since the derivative of theHeaviside unit step function is the Dirac δ-function [31], thederivative of this idealized signal would be a “pulse train” that is alinear combination of Dirac δ-functions. This pulse train would containall the information encoded in the discrete-level signal, and it wouldhave infinitely wide bandwidth and infinitely large power. Both thebandwidth and the power may then be reduced to the desired levels byfiltering the pulse train with a lowpass filter. If the time-bandwidthproduct (TBP) of the filter is sufficiently small so that the pulses inthe filtered pulse train do not overlap, these pulses would stillcontain all the intended information.

On the one hand, converting a narrow-band signal into a wideband pulsetrain has an apparent appeal of no need for “de-spreading”: One maysimply obtain samples at the peaks of the pulses to obtain all theinformation encoded in the signal. On the other hand, at first glancesuch a pulse train is not suitable for practical communication systems,and especially for covert communications. Indeed, let us consider apulse train with a given average pulse rate and power. The average PSDof this train could be made arbitrary small, since it is inverselyproportional to the bandwidth. However, the peak-to-average power ratio(PAPR) of such a train would be proportional to the bandwidth, makingthe wideband signal extremely impulsive (super-Gaussian). First, suchhigh crest factor of the pulse train puts a serious burden on thetransmitter hardware, potentially making this burden prohibitive (e.g.for PAPR > 30 dB). Secondly, the high-PAPR structure of a pulse trainmakes it easily detectable by simple thresholding in the time domain,seemingly making it unsuitable for covert communications. Thirdly, itmay appear that sharing the wideband channel by multiple users wouldrequire explicit allocation of the pulse arrival times for eachsub-channel, which would be impractical in most cases.

Further objects and features of the present invention will becomeapparent to the ones skilled in the art upon examination of thefollowing description and the accompanying drawings. It is intended thatany additional objects and features be incorporated herein.

Time Domain Analysis of 1st- and 2nd-Order Delta-Sigma (Δ∑) ADCs WithLinear Analog Loop Filters

Nowadays, delta-sigma (Δ∑) ADCs are used for converting analog signalsover a wide range of frequencies, from DC to several megahertz. Theseconverters comprise a highly oversampling modulator followed by adigital/decimation filter that together produce a high-resolutiondigital output [40-42]. As discussed in this section, which reviews thebasic principle of operation of ΔΣ ADCs from a time domain prospective,a sample of the digital output of a ΔΣ ADC represents its continuous(analog) input by a weighted average over a discrete time interval (thatshould be smaller than the inverted Nyquist rate) around that sample.

Since frequency domain representation is of limited use in analysis ofnonlinear systems, let us first describe the basic ΔΣ ADCs with 1st- and2nd-order linear analog loop filters in the time domain. Such 1st- and2nd-order ΔΣ ADCs are illustrated in panels I and II of FIG. 1 ,respectively. Note that the vertical scales of the shown fragments ofthe signal traces vary for different fragments.

Without loss of generality, we may assume that if the input D to theflip-flop is greater than zero, D > 0, at a specific instance in theclock cycle (e.g. the rising edge), then the output Q takes a negativevalue Q = -V_(c). If D < 0 at a rising edge of the clock, then theoutput Q takes a positive value Q = V_(c). At other times, the output Qdoes not change. We also assume in this example that x(t) is effectivelyband-limited, and is bounded by V_(c) so that |x(t)| < V_(c) for all t.Further, the clock frequency F_(s) is significantly higher (e.g. by morethan about 2 orders of magnitude) than the bandwidth B_(x) of x(t),log₁₀(F_(s)/B_(x)) ≳ 2. It may be then shown that, with the aboveassumptions, the input D to the flip-flop would be a zero-mean signalwith an average zero crossing rate much higher than the bandwidth ofx(t).

Note that in the limit of infinitely large clock frequency F_(s) (F_(s)→ ∞) the behavior of the flip-flop would be equivalent to that of ananalog comparator. Thus, while in practice a finite flip-flop clockfrequency is used, based on the fact that it is orders of magnitudelarger that the bandwidth of the signal of interest we may usecontinuous-time (e.g. (w _(*) y)(t) and x(t - Δt)) rather thandiscrete-time (e.g. (w _(*) y)[k] and x[k - m]) notations in referenceto the ADC outputs, as a shorthand to simplify the mathematicaldescription of our approach.

As can be seen in FIG. 1 , for the 1st-order modulator shown in panel I

$\begin{matrix}{\overline{x(t)-y(t)} = 0,} & \text{­­­(1)}\end{matrix}$

and for the 2nd-order modulator shown in panel II

$\begin{matrix}{\overline{\overset{˙}{y}(t)} = \frac{1}{\tau}\left\lbrack \overline{x(t)-y(t)} \right\rbrack\mspace{6mu},} & \text{­­­(2)}\end{matrix}$

where the overdot denotes a time derivative, and the overlines denoteaveraging over a time interval between any pair of threshold (includingzero) crossings of D (such as, e.g., the interval ΔT shown in FIG. 1 ).Indeed, for a continuous function f(t), the time derivative of itsaverage over a time interval ΔT may be expressed as

$\begin{matrix}{\overline{\overset{˙}{f}(t)} = \overset{.}{\overline{f(t)}} = \frac{\text{d}}{\text{d}t}\left\lbrack {\frac{1}{\Delta T}{\int_{t - \Delta T}^{t}{\text{d}s\mspace{6mu} f(s)}}} \right\rbrack = \frac{1}{\Delta T}\left\lbrack {f(t) - f\left( {t - \Delta T} \right)} \right\rbrack,} & \text{­­­(3)}\end{matrix}$

and it will be zero if f (t) - f (t-ΔT) = 0.

Now, if the time averaging is performed by a lowpass filter with animpulse response w(t) and a bandwidth B_(w) much smaller than the clockfrequency, B_(w) << F_(s), equation (1) implies that the filtered outputof the 1st-order ΔΣ ADC would be effectively equal to the filteredinput,

$\begin{matrix}{\left( {w \ast y} \right)(t) = \left( {w \ast x} \right)(t) + \delta y,} & \text{­­­(4)}\end{matrix}$

where the asterisk denotes convolution, and the term δy (the “ripple”,or “digitization noise”) is small and will further be neglected. Wewould assume from here on that the filter w(t) has a flat frequencyresponse and a constant group delay Δt over the bandwidth of x(t). Thenequation (4) may be rewritten as

$\begin{matrix}{\left( {w \ast y} \right)(t) = x\left( {t - \Delta t} \right),} & \text{­­­(5)}\end{matrix}$

and the filtered output would accurately represent the input signal.

Since y(t) is a two-level staircase signal with a discrete step durationn/F_(s), where n is a natural (counting) number, it may be accuratelyrepresented by a 1-bit discrete sequence y[k] with the sampling rateF_(s). Thus the subsequent conversion to the discrete (digital) domainrepresentation of x(t) (including the convolution of y[k] with w[k] anddecimation to reduce the sampling rate) is rather straightforward andwill not be discussed further.

If the input to a 1st-order ΔΣ ADC consists of a signal of interest x(t)and an additive noise n(t), then the filtered output may be written as

$\begin{matrix}{\left( {w \ast y} \right)(t) = x\left( {t - \Delta t} \right) + \left( {w \ast v} \right)(t),} & \text{­­­(6)}\end{matrix}$

provided that |x(t - Δt) + (w _(*) v)(t)| < V_(c) for all t. Since w(t)has a flat frequency response over the bandwidth of x(t), it would notchange the power spectral density of the additive noise v(t) in thesignal passband, and the only improvement in the passbandsignal-to-noise ratio for the output (w _(*) y)(t) would come from thereduction of the quantization noise δy by a well designed filter w(t).

Similarly, equation (2) implies that the filtered output of the2nd-order ΔΣ ADC would be effectively equal to the filtered inputfurther filtered by a 1st order lowpass filter with the time constant τand the impulse response h_(τ)(t),

$\begin{matrix}{\left( {w \ast y} \right)(t) = \left( {h_{\tau} \ast w \ast x} \right)(t).} & \text{­­­(7)}\end{matrix}$

From the differential equation for a 1st order lowpass filter it followsthat h_(τ) _(*)(w + τẇ) = w, and thus we may rewrite equation (7) as

$\begin{matrix}{\left( {h_{\tau} \ast \left( {w + \tau\overset{˙}{w}} \right) \ast y} \right)(t) = \left( {h_{\tau} \ast w \ast x} \right)(t).} & \text{­­­(8)}\end{matrix}$

Provided that τ is sufficiently small (e.g., τ ≲ 1/(4πB_(x))), equation(8) may be further rewritten as

$\begin{matrix}{\left( {\left( {w + \tau\overset{˙}{w}} \right) \ast y} \right)(t) = \left( {w \ast x} \right)(t) = x\left( {t - \Delta t} \right).} & \text{­­­(9)}\end{matrix}$

The effect of the 2nd-order loop filter on the quantization noise δy isoutside the scope of this disclosure and will not be discussed.

SUMMARY

Since at any given frequency a linear filter affects both the noise andthe signal of interest proportionally, when a linear filter is used tosuppress the interference outside of the passband of interest theresulting signal quality is affected only by the total power andspectral composition, but not by the type of the amplitude distributionof the interfering signal. Thus a linear filter cannot improve thepassband signal-to-noise ratio, regardless of the type of noise. On theother hand, a nonlinear filter has the ability to disproportionatelyaffect signals with different temporal and/or amplitude structures, andit may reduce the spectral density of non-Gaussian (e.g. impulsive)interferences in the signal passband without significantly affecting thesignal of interest. As a result, the signal quality may be improved inexcess of that achievable by a linear filter. Such non-Gaussian (and, inparticular, impulsive, or outlier, or transient) noise may originatefrom a multitude of natural and technogenic (man-made) phenomena. Thetechnogenic noise specifically is a ubiquitous and growing source ofharmful interference affecting communication and data acquisitionsystems, and such noise may dominate over the thermal noise. While thenon-Gaussian nature of technogenic noise provides an opportunity for itseffective mitigation by nonlinear filtering, current state-of-the-artapproaches employ such filtering in the digital domain, afteranalog-to-digital conversion. In the process of such conversion, thesignal bandwidth is reduced, and the broadband non-Gaussian noise maybecome more Gaussian-like. This substantially diminishes theeffectiveness of the subsequent noise removal techniques.

The present invention overcomes the limitations of the prior art throughincorporation of a particular type of nonlinear noise filtering of theanalog input signal into nonlinear analog filters preceding an ADC,and/or into loop filters of ΔΣ ADCs. Such ADCs thus combineanalog-to-digital conversion with analog nonlinear filtering, enablingmitigation of various types of in-band non-Gaussian noise andinterference, especially that of technogenic origin, including broadbandimpulsive interference. This may considerably increase quality of theacquired signal over that achievable by linear filtering in the presenceof such interference. An important property of the presented approach isthat, while being nonlinear in general, the proposed filters wouldlargely behave linearly. They would exhibit nonlinear behavior onlyintermittently, in response to noise outliers, thus avoiding thedetrimental effects, such as instabilities and intermodulationdistortions, often associated with nonlinear filtering.

The intermittently nonlinear filters of the present invention would alsoenable separation of signals (and/or signal components) withsufficiently different temporal and/or amplitude structures in the timedomain, even when these signals completely or partially overlap in thefrequency domain. In addition, such separation may be achieved withoutreducing the bandwidths of said signal components.

Even though the nonlinear filters of the present invention areconceptually analog filters, they may be easily implemented digitally,for example, in Field Programmable Gate Arrays (FPGAs) or software. Suchdigital implementations would require very little memory and would betypically inexpensive computationally, which would make them suitablefor real-time signal processing.

To meet the undetectability requirement, in a steganographic system thestego signals should be statistically indistinguishable from the coversignals. For physical layer transmissions, undetectability may beenhanced by requiring that the payload and the cover have the samebandwidth and spectral content, the same apparent temporal and amplitudestructures, and that there are no explicit differences in the spectraland/or temporal allocations for the cover signals and the payloadmessages. For a mixture of such signals, neither linear nor nonlinearfiltering alone can separate the signals. Favorably, however, linearfiltering may significantly, and differently, affect the temporal andamplitude structure of many natural and the majority of technogenic(man-made) signals. For example, such filtering can often convert theamplitude distribution of a pulse train from super-Gaussian intoapparently Gaussian and/or sub-Gaussian, and vice versa. On the otherhand, a nonlinear filter is capable of disproportionately affectingspectral densities of signals with distinct temporal and/or amplitudestructures even when the signals have the same spectral content.Therefore, in the present invention a proper synergistic combination oflinear and nonlinear filtering is employed to effectively separate such“indistinguishable” cover and stego signals.

Another object of the present invention is data communications and, inparticular, communicating over longer distances at lower power andenergy dissipation. For example, in low-power wide-area networks(LPWANs), various trade-offs among the bandwidth, data rates, and energyper bit have different effects on the quality of service under differentpropagation conditions (e.g. fading and multipath), interferencescenarios, multi-user requirements, and design constraints. Suchcompromises, and the manner in which they are implemented, furtheraffect other technical aspects, such as system’s computationalcomplexity and power efficiency. At the same time, this difference intrade-offs also adds to the technical flexibility in addressing abroader range of communications applications, both static and mobile. Inthe communications method and apparatus of the present invention thecontrol of the quality of service is performed through the change in thespectral efficiency (i.e., the data rate at a given bandwidth), and/orthrough changing the energy per bit as an additional trade-offparameter.

Further scope and the applicability of the invention will be clarifiedthrough the detailed description given hereinafter. It should beunderstood, however, that the specific examples, while indicatingpreferred embodiments of the invention, are presented for illustrationonly. Various changes and modifications within the spirit and scope ofthe invention should become apparent to those skilled in the art fromthis detailed description. Furthermore, all the mathematicalexpressions, diagrams, and the examples of hardware implementations areused only as a descriptive language to convey the inventive ideasclearly, and are not limitative of the claimed invention.

BRIEF DESCRIPTION OF FIGURES

FIG. 1 . ΔΣ ADCs with 1st-order (I) and 2nd-order (II) linear loopfilters.

FIG. 2 . Simplified diagram of improving receiver performance in thepresence of impulsive interference.

FIG. 3 . Illustrative ABAINF block diagram.

FIG. 4 . Illustrative examples of the transparency functions and theirrespective influence functions.

FIG. 5 . Block diagrams of CMTFs with blanking ranges [α_, α₊] (a) and[V₋, V₊]/G (b).

FIG. 6 . Resistance of CMTF to outlier noise. The cross-hatched timeintervals in panel (c) correspond to nonlinear CMTF behavior (zero rateof change).

FIG. 7 . Illustration of differences in the error signal for the exampleof FIG. 6 . The cross-hatched time intervals indicate nonlinear CMTFbehavior (zero rate of change).

FIG. 8 . Simplified illustrated schematic of CMTF circuitimplementation.

FIG. 9 . Resistance of the CMTF circuit of FIG. 8 to outlier noise. Thecross-hatched time intervals in the lower panel correspond to nonlinearCMTF behavior.

FIG. 10 . Using sums and/or differences of input and output of CMTF andits various intermediate signals for separating impulsive (outlier) andnon-impulsive signal components.

FIG. 11 . Illustration of using CMTF with appropriate blanking range forseparating impulsive and non-impulsive (“background”) signal components.

FIG. 12 . Illustrative block diagrams of an ADiC with time parameter τand blanking range [α_, α₊].

FIG. 13 . Simplified illustrative electronic circuit diagram of usingCMTF with appropriately chosen blanking range [α_, α₊] for separatingincoming signal x(t) into impulsive i(t) and non-impulsive s(t)(“background”) signal components.

FIG. 14 . Illustration of separating incoming signal x(t) into impulsivei(t) and non-impulsive s(t) (“background”) components by the circuit ofFIG. 13 .

FIG. 15 . Illustration of separation of discrete input signal “x” intoimpulsive component “aux” and non-impulsive (“background”) component“prime” using the MATLAB function of §2.5 with appropriately chosenblanking values “alpha_p” and “alpha_m”.

FIG. 16 . Illustrative block diagram of a circuit implementing equation(21) and thus tracking a q th quantile of y(t).

FIG. 17 . Illustration of MTF convergence to steady state for differentinitial conditions.

FIG. 18 . Illustration of QTFs′ convergence to steady state fordifferent initial conditions.

FIG. 19 . Illustration of separation of discrete input signal “x” intoimpulsive component “aux” and non-impulsive (“background”) component“prime” using the MATLAB function of §3.3 with the blanking rangecomputed as Tukey’s range using digital QTFs.

FIG. 20 . Transparency function described by equation (30).

FIG. 21 . Illustrative block diagram of an adaptive intermittentlynonlinear filter for mitigation of outlier noise in the process ofanalog-to-digital conversion.

FIG. 22 . Equivalent block diagram for the filter shown in FIG. 21operating in linear regime.

FIG. 23 . Impulse and frequency responses of w[k] and w[k] + τẇ[k] usedin the subsequent examples.

FIG. 24 . Comparison of simulated channel capacities for the linearprocessing chain (solid curves) and the CMTF-based chains with β = 3(dotted and dashed curves). The dashed curves correspond to channelcapacities for the CMTF-based chain with added interference in anadjacent channel. The asterisks correspond to the noise and adjacentchannel interference conditions used in FIG. 25 .

FIG. 25 . Illustration of changes in the signal time- and frequencydomain properties, and in its amplitude distributions, while itpropagates through the signal processing chains, linear (points (a),(b), and (c) in panel II of FIG. 22 ), and the CMTF-based (points Ithrough IV, and point V, in FIG. 21 ).

FIG. 26 . Alternative topology for signal processing chain shown in FIG.21 .

FIG. 27 . ΔΣ ADC with an CMTF-based loop filter.

FIG. 28 . Modifying the amplitude density of the difference signal x - yby a 1st order lowpass filter.

FIG. 29 . Impulse and frequency responses of w[k] and w[k] +(4πB_(x))⁻¹ẇ[k] used in the examples of FIG. 30 .

FIG. 30 . Comparative performance of ΔΣ ADCs with linear and nonlinearanalog loop filters.

FIG. 31 . Resistance of ΔΣ ADC with CMTF-based loop filter to increasein impulsive noise.

FIG. 32 . Outline of ΔΣ ADC with adaptive CMTF-based loop filter.

FIG. 33 . Comparison of simulated channel capacities for the linearprocessing chain (solid lines) and the CMTF-based chains with β = 1.5(dotted lines). The meaning of the asterisks is explained in the text.

FIG. 34 . Reduction of the spectral density of impulsive noise in thesignal baseband without affecting that of the signal of interest.

FIG. 35 . Reduction of the spectral density of impulsive noise in thesignal baseband without affecting that of the signal of interest.(Illustration similar to FIG. 34 with additional interference in anadjacent channel.)

FIG. 36 . Illustrative signal chains for a ΔΣ ADC with linear loop anddecimation filters (panel (a)), and for a ΔΣ ADC with linear loop filterand ADiC-based digital filtering (panel (b)).

FIG. 37 . Illustrative time-domain traces at points I through VI of FIG.36 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise).

FIG. 38 . Illustrative signal chains for a ΔΣ ADC with linear loop anddecimation filters (panel (a)), and for a ΔΣ ADC with linear loop filterand CMTF-based digital filtering (panel (b)).

FIG. 39 . Illustrative time-domain traces at points I through VI of FIG.38 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise).

FIG. 40 . Illustrative signal chains for a ΔΣ ADC with linear loop anddecimation filters (panel (a)), and for a ΔΣ ADC with linear loop filterand ADiC-based digital filtering (panel (b)), with additional clippingof the analog input signal.

FIG. 41 . Illustrative time-domain traces at points I through VI of FIG.40 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise).

FIG. 42 . Illustrative signal chains for a ΔΣ ADC with linear loop anddecimation filters (panel (a)), and for a ΔΣ ADC with linear loop filterand CMTF-based digital filtering (panel (b)), with additional clippingof the analog input signal.

FIG. 43 . Illustrative time-domain traces at points I through VI of FIG.42 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise).

FIG. 44 . Alternative ADiC structure.

FIG. 45 . Illustrative examples of differential influence functions andtheir respective difference responses.

FIG. 46 . Alternative ADiC structure with a boxcar depreciator.

FIG. 47 . Illustrative signal traces for the ADiC shown in FIG. 46 withthe DCL established by a linear lowpass filter.

FIG. 48 . Illustrative signal traces for the ADiC shown in FIG. 46 withthe DCL established by a TTF.

FIG. 49 . Illustrative signal traces for the ADiC shown in FIG. 46 withthe DCL established by a TTF (continued).

FIG. 50 . Simplified ADiC structure.

FIG. 51 . Example of a simplified ADiC structure with a differentialblanker as a depreciator.

FIG. 52 . Illustrative electronic circuit for the ADiC structure shownin FIG. 51 .

FIG. 53 . Illustrative signal traces for the ADiC shown in FIG. 52(LTspice simulation).

FIG. 54 . Example of applying a numerical version of an ADiC describedin Section 8 to the input signal used in FIG. 15 .

FIG. 55 . Two cascaded ADiCs (panel (a)), and an alternative cascadedADiC structure (panel (b)).

FIG. 56 . Illustrative time-domain traces for the ADiC structures shownin FIG. 55 .

FIG. 57 . Illustrative block diagram of a complex-valued CMTF.

FIG. 58 . Illustrative use of a complex-valued ADiC for interferencemitigation in a quadrature receiver (QPSK-modulated signal).

FIG. 59 . Example of complex-valued ADiC structure.

FIG. 60 . Example of degrading signal of interest by removing “outliers”instead of “outlier noise”.

FIG. 61 . Illustration of “excess band” observation of outlier noise.

FIG. 62 . Illustrative examples of excess band responses.

FIG. 63 . Illustrative example of spectral inversion by ADiC.

FIG. 64 . Illustrative example of spectral “cockroach effect” caused byoutlier removal.

FIG. 65 . Illustration of complementary ADiC filtering (CAF) structurefor ADiC-based outlier noise mitigation for passband signal.

FIG. 66 . Complementary ADiC filter (CAF) for removing wideband noiseoutliers while preserving band-limited signal of interest.

FIG. 67 . CAF block diagram.

FIG. 68 . Analog (panel (a)) and digital (panel (b)) ABAINF deploymentfor mitigation of non-Gaussian (e.g. outlier) noise in the process ofanalog-to-digital conversion.

FIG. 69 . Analog (panel (a)) and digital (panel (b)) ABAINF-basedoutlier filtering in ΔΣ ADCs.

FIG. 70 . Example of a ΔΣ ADC with an ADIC-based decimation filter forenhanced interference mitigation.

FIG. 71 . Example of using a 1st order highpass filter prior to ADiC forenhanced interference mitigation.

FIG. 72 . Impulse and frequency responses of w[k] and w[k] + Δw[k] usedin the example of FIG. 71 .

FIG. 73 . Illustration of spectral reshaping of impulsive noise by anADiC.

FIG. 74 . Example of using ADiC-based filtering for mitigation ofimpulsive noise in the presence of strong adjacent channel interference.

FIG. 75 . Two signal processing chains for the example described in§11.5.

FIG. 76 . Examples of the time-domain traces and the PSDs of the signalsat points I, II, III, IV, and V in FIG. 75 .

FIG. 77 . Example of a ΔΣ ADC with an ADIC-based decimation filter formitigation of wideband impulsive noise affecting the baseband signal ofinterest in the presence of a strong adjacent-channel interference.

FIG. 78 . Example of using ADiC-based filtering in direct conversionreceiver architecture with quadrature baseband ADCs.

FIG. 79 . Example of using ADiC-based filtering in superheterodynereceivers.

FIG. 80 . Example of a conceptual schematic of a sub-circuit for anOTA-based implementation of a depreciator with the transparency functiongiven by equation (30) and depicted in FIG. 20 .

FIG. 81 . Example of an OTA-based squaring circuit (e.g. “SQ” circuit inFIG. 59 ) for a complex-valued signal.

FIG. 82 . Example of a conceptual schematic of a sub-circuit for anOTA-based implementation of a depreciator with the transparency functiondepicted in FIGS. 57 and 59 and given by equation (62).

FIG. 83 . Illustration of pileup effect: When “width” of pulses becomesgreater than distance between them, pulses may begin to overlap andinterfere with each other. For pulses with same spectral content, PSDsof pulse sequences are identical, yet their temporal and amplitudestructures are substantially different.

FIG. 84 . Pairs of matched filters with different time-bandwidthproducts, but same frequency responses and same “combined” impulseresponse. In this example, w(t) is root-raised-cosine filter, and thus(w_(*)w)(t) is raised-cosine filter.

FIG. 85 . Example of FIG. 84 extended to two dimensions.

FIG. 86 . Using pileup effect for obfuscation of temporal and amplitudestructure of transmitted signal. In transmitter, filtering withlarge-TBP filter reduces crest factor of pulse train, making it appearas Gaussian or sub-Gaussian. In receiver, filtering with matched filterrestores signal’s distinct temporal and amplitude structure.

FIG. 87 . Relations among rate, PAPR, and SNR in pulse train used forlow-SNR communications. For TBP = 1 and 10⁻² ≤ ε ≤ 10⁻³, “raw” ratelimits for detectible pulses of equal magnitudes vary from few percent(for pulse counting) to about half of Shannon rate (for synchronouspulse detection).

FIG. 88 . Intermittently Nonlinear Filtering (INF): Outliers areidentified as protrusions outside of fenced range, and their values arereplaced by those in mid-range. Otherwise, signal is not affected.“Auxiliary” output is difference between input and “prime” INF output.

FIG. 89 . For low pulse rates

$\left( {\text{e}\text{.g}\mathcal{R} \ll \frac{1}{2}{{\Delta B}/\text{TBP}}} \right),$

IQR provides reliable measure of additive Gaussian noise power, σ_(n) ∝IQR. Root-raised-cosine pulses (for which

ℛ₀ = (4T_(s))⁻¹

) are used in this example.

FIG. 90 . Overall behavior of QTF fencing is similar to that with“exact” quartile filters in moving boxcar window of width ΔT = 2 ×IQR/µ.

FIG. 91 . Simplified diagram of first example (Section 12.5.1).

FIG. 92 . Detailed particular example for basic concept highlighted inFIG. 91 .

FIG. 93 . Simplified diagram of second example (Section 12.5.2).

FIG. 94 . Both high-SNR and low-SNR pulse trains are disguised asGaussian noises with same spectral content. In this example, timeduration of g₁₂(t) is not much larger than average time interval betweenpulses in x₁(t), and thus

x₁^(⋆)(t)

is slightly sub-Gaussian.

FIG. 95 . Both high-SNR and low-SNR pulse trains are recovered inreceiver. First INF accomplishes both recovery of high-SNR pulse trainand its removal from mixture.

FIG. 96 . Impulse responses of filters used in FIGS. 94 and 95 , andtheir convolutions.

FIG. 97 . Basic concept of “friendly in-band jamming.”

FIG. 98 . OFDM PAPR reduction by large-TBP filter.

FIG. 99 . Friendly in-band jamming of OFDM signal. Combination of linearand nonlinear filtering in receiver is used for effective separation ofOFDM and “friendly jamming” signals, although both signals in receivedmixture have effectively same spectral characteristics and temporal andamplitude structures, and there are no explicit differences in theirtemporal allocations.

FIG. 100 . PAPR(N_(p)) ≈ 1.143 N_(p)/N_(s) for N_(p)/N_(s) >> 1 for RCpulses with β=½.

FIG. 101 . AWGN SNR limits for different BER as functions of samplesbetween pulses for raised-cosine pulses with β = ½ and N_(s) = 2.

FIG. 102 . Illustration of synchronization procedure described by (74)through (77). AWGN SNR = -20 dB is chosen to be low, and M = 32respectively high, to emphasize robustness even when BER ≈ ⅓.

FIG. 103 . Calculated and simulated BERs as functions of AWGN SNRs forN_(p) = 32 and N_(p) = 256. For shown SNR ranges, MPA function with M=8provides reliable synchronization. (Compare with SNR limits in FIG. 101.)

FIG. 104 . For BER smaller than about 10⁻¹, less computationallyexpensive modulo magnitude averaging (e.g. given by (78)) can be usedfor synchronization. Modulo power averaging (with “extra point,” e.g.given by (74)) should be used when reliable synchronization for full BERrange is desired.

FIG. 105 . Transmitter waveform is constructed as sum of scaled andtime-shifted/delayed large-TBP pulses. In receiver, IIR allpass filteris used to recover small-TBP pulse train.

FIG. 106 . Comparative illustration of PAPR and K_(dBG) as measures ofpeakedness for pulse trains.

FIG. 107 . Using pulse trains for low-SNR communications: Large-TBPpulse shaping (i) “hides” pulse train, obscuring its temporal andamplitude structure, and (ii) reduces its PAPR, making signal suitablefor transmission. In receiver, pulse train is restored by matchedlarge-TBP filtering. High PAPR of restored pulse train enables low-SNRmessaging. To make link more robust to outlier interference and toincrease apparent SNR, analog-to-digital conversion in receiver may becombined with intermittently nonlinear filtering.

FIG. 108 . Illustration of transmitter for single-sideband M-ary ASPMlink with constant-envelope pulses and M = 8 (three bits per pulse) fornoncoherent detection and M = 16 (four bits per pulse) for coherentdetection.

FIG. 109 . Illustration of noncoherent and coherent receivers forsingle-sideband M-ary ASPM link with M = 8 (three bits per pulse) fornoncoherent detection and M = 16 (four bits per pulse) for coherentdetection.

FIG. 110 . Illustration of transmitter for single-sideband M-ary ASPMlink with constant-envelope pulses and M = 32 (five bits per pulse) fornoncoherent detection and M = 64 (six bits per pulse) for coherentdetection.

FIG. 111 . Illustration of noncoherent and coherent receivers forsingle-sideband M-ary ASPM link with M = 32 (five bits per pulse) fornoncoherent detection and M = 64 (six bits per pulse) for coherentdetection.

FIG. 112 . Uncoded BER vs. E_(b/)N₀ performances of coherent andnoncoherent M-ASPM in AWGN channel for several values of M.

FIG. 113 . Uncoded BER vs. SNR performance of coherent and noncoherentM-ASPM in AWGN channel for several values of M.

FIG. 114 . Illustration of pulse shaping for single-sidebandconstant-envelope M-ASPM with information encoded in plurality ofequidistant designed pulse trains.

FIG. 115 . Illustration of single-sideband M-ary ASPM link withconstant-envelope pulses and noncoherent detection.

FIG. 116 . For sufficiently different L_(i) and L_(j),cross-correlations of ĝ_(i)[k] and ĝ_(j)[k] for constant-envelope PSFsgiven by (111) have large TBPs.

ABBREVIATIONS

ABAINF: Analog Blind Adaptive Intermittently Nonlinear Filter; ACF:autocorrelation function; A/D: Analog-to-Digital; ADC: Analog-to-DigitalConverter (or Conversion); ADiC: Analog Differential Clipper; AFE:Analog Front End; AGC: Automatic Gain Control; ASIC:Application-Specific Integrated Circuit; ASPM: Aggregate Spread PulseModulation; ASSP: Application-Specific Standard Product; AWGN: AdditiveWhite Gaussian Noise;

BAINF: Blind Adaptive Intermittently Nonlinear Filter; BER: Bit ErrorRate, or Bit Error Ratio; BPSK: Binary Phase-Shift Keying;

CAF: Complementary ADiC Filter (or Filtering); CDL: CanonicalDifferential Limiter; CDMA: Code Division Multiple Access; CINF:Complementary Intermittently Nonlinear Filter (or Filtering); CLT:Central Limit Theorem; CMTF: Clipped Mean Tracking Filter; COTS:Commercial Off-The-Shelf; CPD: Coincidence Pulse Detection;

D/A: Digital-to-Analog; DAC: Digital-to-Analog Converter (orConversion); DCL: Differential Clipping Level; DELDC: Dual Edge LimitDetector Circuit; DFT: Discrete Fourier Transform; DSP: Digital SignalProcessing/Processor;

EMC: electromagnetic compatibility; EMI: electromagnetic interference;ENBW: equivalent noise bandwidth;

FFT: Fast Fourier Transform; FIR: Finite Impulse Response; FPGA: FieldProgrammable Gate Array;

HSDPA: High Speed Downlink Packet Access;

IC: Integrated Circuit; IF: Intermediate Frequency; IDFT: InverseDiscrete Fourier Transform; INF: Intermittently Nonlinear Filter (orFiltering); i.i.d.: Independent and Identically Distributed; IoT:Internet of Things; IpI: Interpulse Interval; I/Q: In-phase/Quadrature;IQR: interquartile range;

LNA: Low-Noise Amplifier; LO: Local Oscillator; LoRa: Long Range(proprietary LPWAN modulation technique); LPI:Low-Probability-of-Intercept; LPWAN: Low-Power Wide-Area Network;

MAD: Mean/Median Absolute Deviation; M-ASPM: M-ary Aggregate SpreadPulse Modulation; MATLAB: MATrix LABoratory (numerical computingenvironment and fourth-generation programming language developed byMathWorks); MCA: Modulo Count Averaging; MCT: Measure of CentralTendency; MMA: Modulo Magnitude Averaging; MOS:Metal-Oxide-Semiconductor; MPA: Modulo Power Averaging; MTF: MedianTracking Filter;

NDL: Nonlinear Differential Limiter;

OOB: Out-Of-Band; ORB: Outlier-Removing Buffer; OTA: OperationalTransconductance Amplifier;

PA: Power Amplifier; PAPR: Peak-to-Average Power Ratio; PDF: ProbabilityDensity Function; PHY: physical layer; PSD: Power Spectral Density; PSF:Pulse Shaping Filter;

QTF: Quartile (or Quantile) Tracking Filter;

RC: Raised-Cosine; RF: Radio Frequency; RFI: Radio FrequencyInterference; RMS: Root Mean Square; RRC: Robust Range Circuit; RRC:Root Raised Cosine; RX: Receiver;

SF: Spreading Factor (for LoRa); SIR: Signal-to-Interference Ratio;SINR: Signal-to-Interference-plus-Noise Ratio; SNR: Signal-to-NoiseRatio; SCS: Switch Control Signal; SPDT: Single Pole Double-Throwswitch; SRRC: Square-Root-Raised-Cosine;

TBP: Time-Bandwidth Product; ToA: Time-on-Air; TTF: Trimean TrackingFilter; TX: Transmitter; UWB: Ultra-wideband;

WCC: Window Comparator Circuit; WDC: Window Detector Circuit; WMCT:Windowed Measure of Central Tendency; WML: Windowed Measure of Location;

VGA: Variable-Gain Amplifier

DETAILED DESCRIPTION

As required, detailed embodiments of the present invention are disclosedherein. However, it is to be understood that the disclosed embodimentsare merely exemplary of the invention that may be embodied in variousand alternative forms. The figures are not necessarily to scale; somefeatures may be exaggerated or minimized to show details of particularcomponents. Therefore, specific structural and functional detailsdisclosed herein are not to be interpreted as limiting, but merely as arepresentative basis for the claims and/or as a representative basis forteaching one skilled in the art to variously employ the presentinvention.

Moreover, except where otherwise expressly indicated, all numericalquantities in this description and in the claims are to be understood asmodified by the word “about” in describing the broader scope of thisinvention. Practice within the numerical limits stated is generallypreferred. Also, unless expressly stated to the contrary, thedescription of a group or class of components as suitable or preferredfor a given purpose in connection with the invention implies thatmixtures or combinations of any two or more members of the group orclass may be equally suitable or preferred.

It should be understood that the word “analog”, when used in referenceto various embodiments of the invention, is used only as a descriptivelanguage to convey the inventive ideas clearly, and is not limitative ofthe claimed invention. Specifically, the word “analog” mainly refers tousing differential and/or integral equations (and thus suchanalog-domain operations as differentiation, antidifferentiation, andconvolution) in describing various signal processing structures andtopologies of the invention. In reference to numerical or digitalimplementations of the disclosed analog structures, it is to beunderstood that such numerical or digital implementations simplyrepresent finite-difference approximations of the respective analogoperations and thus may be accomplished in a variety of alternativeways.

For example, a “numerical derivative” of a quantity x(t) sampled atdiscrete time instances t_(k) such that t_(k+1) = t_(k) + dt should beunderstood as a finite difference expression approximating a “true”derivative of x(t). One skilled in the art will recognize that thereexist many such expressions and algorithms for estimating the derivativeof a mathematical function or function subroutine using discrete sampledvalues of the function and perhaps other knowledge about the function.However, for sufficiently high sampling rates, for digitalimplementations of the analog structures described in this disclosuresimple two-point numerical derivative expressions may be used. Forexample, a numerical derivative of x(t_(k)) may be obtained using thefollowing expressions:

$\begin{matrix}{\begin{array}{l}{\overset{˙}{x}\left( t_{k} \right) = \frac{x\left( t_{k + 1} \right) - x\left( t_{k} \right)}{\text{d}t},} \\{\overset{˙}{x}\left( t_{k} \right) = \frac{x\left( t_{k} \right) - x\left( t_{k - 1} \right)}{\text{d}t},} \\{\overset{˙}{x}\left( t_{k} \right) = \frac{x\left( t_{k + 1} \right) - x\left( t_{k - 1} \right)}{2\,\text{d}t}.}\end{array}\mspace{6mu}\text{or}} & \text{­­­(10)}\end{matrix}$

Further, the quantities proportional to numerical derivatives may beobtained using the following expressions:

$\begin{matrix}{\begin{array}{l}{\overset{˙}{x}\left( t_{k} \right) \propto x\left( t_{k + 1} \right) - x\left( t_{k} \right),} \\{\overset{˙}{x}\left( t_{k} \right) \propto x\left( t_{k} \right) - x\left( t_{k - 1} \right),} \\{\overset{˙}{x}\left( t_{k} \right) \propto x\left( t_{k + 1} \right) - x\left( t_{k - 1} \right).}\end{array}\text{or}} & \text{­­­(11)}\end{matrix}$

The detailed description of the invention is organized as follows.

Section 1 (“Analog Intermittently Nonlinear Filters for Mitigation ofOutlier Noise”) outlines the general idea of employing intermittentlynonlinear filters for mitigation of outlier (e.g. impulsive) noise, andthus improving the performance of a communications receiver in thepresence of such noise. E.g., §1.1 (“Motivation and simplified systemmodel”) describes a simplified diagram of improving receiver performancein the presence of impulsive interference.

Section 2 (“Analog Blind Adaptive Intermittently Nonlinear Filters(ABAINFs) with the desired behavior”) introduces a practical approach toconstructing analog nonlinear filters with the general behavior outlinedin Section 1, and §2.1 (“A particular ABAINF example”) provides aparticular ABAINF example. Another particular ABAINF example, with theinfluence function of a type shown in panel (iii) of FIG. 4 , is givenin §2.2 (“Clipped Mean Tracking Filter (CMTF)”), and §2.3 (“IllustrativeCMTF circuit”) provides a simplified illustration of implementing a CMTFby solving equation (17) in an electronic circuit. Further, §2.4 (“UsingCMTFs for separating impulsive (outlier) and non-impulsive signalcomponents with overlapping frequency spectra: Analog DifferentialClippers (ADiCs)”) introduces an Analog Differential Clipper (ADiC), and§2.5 (“Numerical implementations of ABAINFs/CMTFs/ADiCs”) provides anexample of a numerical ADiC algorithm and outlines its hardwareimplementation.

Section 3 (“Quantile tracking filters as robust means to establish theABAINF transparency range(s)”) introduces quantile tracking filters thatmay be employed as robust means to establish the ABAINF transparencyrange(s), with §3.1 (“Median Tracking Filter”) discussing the trackingfilter for the 2nd quartile (median), and §3.2 (“Quartile TrackingFilters”) describing the tracking filters for the 1st and 3rd quartiles.Further, §3.3 (“Numerical implementations of ABAINFs/CMTFs/ADiCs usingquantile tracking filters as robust means to establish the transparencyrange”) provides an illustration of using numerical implementations ofquantile tracking filters as robust means to establish the transparencyrange in digital embodiments of ABAINFs/CMTFs/ADiCs, and §3.4 (“Adaptiveinfluence function design”) comments on an adaptive approach toconstructing ADiC influence functions.

Section 4 (“Adaptive intermittently nonlinear analog filters formitigation of outlier noise in the process of analog-to-digitalconversion”) illustrates analog-domain mitigation of outlier noise inthe process of analog-to-digital (A/D) conversion that may be performedby deploying an ABAINF (for example, a CMTF) ahead of an ADC.

While §4 illustrates mitigation of outlier noise in the process ofanalog-to-digital conversion by ADiCs/CMTFs deployed ahead of an ADC,Section 5 (“Δ∑ ADC with CMTF-based loop filter”) discusses incorporationof CMTF-based outlier noise filtering of the analog input signal intoloop filters of ΔΣ analog-to-digital converters.

While §5 describes CMTF-based outlier noise filtering of the analoginput signal incorporated into loop filters of ΔΣ analog-to-digitalconverters, the high raw sampling rate (e.g. the flip-flop clockfrequency) of a ΔΣ ADC (e.g. two to three orders of magnitude largerthan the bandwidth of the signal of interest) may be used for effectiveABAINF/CMTF/ADiC-based outlier filtering in the digital domain,following a Δ∑ modulator with a linear loop filter. This is discussed inSection 6 (“Δ∑ ADCs with linear loop filters and digital ADiC/CMTFfiltering”).

Section 7 (“ADiC variants”) describes several alternative ADiCstructures, and §7.1 (“Robust filters”) comments of various means toestablish robust local measures of location (e.g. central tendency) thatmay be used to establish ADiC differential clipping levels. Inparticular, §7.1.1 (“Trimean Tracking Filter (TTF)”) describes a TrimeanTracking Filter (TTF) as one of such means.

Section 8 (“Simplified ADiC structure”) and §8.1 (“Cascaded ADiCstructures”) describe simplified ADiC structures that may be a preferredway to implement ADiC-based filtering due to their simplicity androbustness.

Section 9 (“ADiC-based filtering of complex-valued signals”) discussesADiC-based filtering of complex-valued signals.

Section 10 (“Hidden outlier noise and its mitigation”) discusses howout-of-band observation of outlier noise enables its efficient in-bandmitigation (in §10.1 (“‘Outliers’ vs. ‘outlier noise’“) and §10.2(“‘Excess band’ observation for in-band mitigation”)), and describes theComplementary ADiC Filtering (CAF) structure (in §10.3 (“ComplementaryADiC Filter (CAF)”)).

Section 11 (“Explanatory comments and discussion”) provides severalcomments on the disclosure given in Sections 1 through 10, withadditional details discussed in §11.1 (“Mitigation of non-Gaussian (e.g.outlier) noise in the process of analog-to-digital conversion: Analogand digital approaches”), §11.2 (“Comments on ΔΣ modulators”), §11.3(“Comparators, discriminators, clippers, and limiters”), §11.4(“Windowed measures of location”), §11.5 (“Mitigation of non-impulsivenon-Gaussian noise”), and §11.6 (“Clarifying remarks”).

Penultimately, Section 12 (“Utilizing pileup effect and intermittentlynonlinear filtering in synthesis of low-SNR and/or covert andhard-to-intercept communication links”) describes the use of synergisticcombinations of linear and nonlinear filtering of the present inventionin synthesis of low-SNR and/or secure communication links.

Finally, Section 13 (“Communicating over longer distances at lower powerand energy dissipation”) addresses yet another object of the presentinvention, which is data communications and, in particular,communicating over longer distances at lower power and energydissipation.

1 Analog Intermittently Nonlinear Filters for Mitigation of OutlierNoise

In the simplified illustration that follows, our focus is not onproviding precise definitions and rigorous proof of the statements andassumptions, but on outlining the general idea of employingintermittently nonlinear filters for mitigation of outlier (e.g.impulsive) noise, and thus improving the performance of a communicationsreceiver in the presence of such noise.

1.1 Motivation and Simplified System Model

Let us assume that the input noise affecting a baseband signal ofinterest with unit power consists of two additive components: (i) aGaussian component with the power P_(G) in the signal passband, and (ii)an outlier (impulsive) component with the power P_(i) in the signalpassband. Thus if a linear antialiasing filter is used before theanalog-to-digital conversion (ADC), the resulting signal-to-noise ratio(SNR) may be expressed as (P_(G) + P_(i))⁻¹.

For simplicity, let us further assume that the outlier noise is whiteand consists of short (with the characteristic duration much smallerthan the reciprocal of the bandwidth of the signal of interest) randompulses with the average inter-arrival times significantly larger thantheir duration, yet significantly smaller than the reciprocal of thesignal bandwidth. When the bandwidth of such noise is reduced to withinthe baseband by linear filtering, its distribution would be wellapproximated by Gaussian [43]. Thus the observed noise in the basebandmay be considered Gaussian, and we may use the Shannon formula [44] tocalculate the channel capacity.

Let us now assume that we use a nonlinear antialiasing filter such thatit behaves linearly, and affects the signal and noise proportionally,when the baseband power of the impulsive noise is smaller than a certainfraction of that of the Gaussian component, P_(i) ≤ εP_(G) (ε ≥ 0)resulting in the SNR (P_(G) + P_(i))⁻¹. However, when the baseband powerof the impulsive noise increases beyond εP_(G), this filter maintainsits linear behavior with respect to the signal and the Gaussian noisecomponent, while limiting the amplitude of the outlier noise in such away that the contribution of this noise into the baseband remainslimited to εP_(G) < P_(i). Then the resulting baseband SNR would be[(1 + ε)P_(G)]⁻¹ > (P_(G) + P_(i))⁻¹. We may view the observed noise inthe baseband as Gaussian, and use the Shannon formula to calculate thelimit on the channel capacity.

As one may see from this example, by disproportionately affectinghigh-amplitude outlier noise while otherwise preserving linear behavior,such nonlinear antialiasing filter would provide resistance to impulsiveinterference, limiting the effects of the latter, for small ε, to aninsignificant fraction of the Gaussian noise. FIG. 2 illustrates thiswith a simplified diagram of improving receiver performance in thepresence of impulsive interference by employing such analog nonlinearfilter before the ADC. In this illustration, ε = 0.2.

2 Analog Blind Adaptive Intermittently Nonlinear Filters (ABAINFs) Withthe Desired Behavior

The analog nonlinear filters with the behavior outlined in §1.1 may beconstructed using the approach shown in FIG. 3 , which provides anillustrative block diagram of an Analog Blind Adaptive IntermittentlyNonlinear Filter (ABAINF).

In FIG. 3 , the influence function [45]

I_(α⁻)^(α₊)(x)

is represented as

I_(α⁻)^(α₊)(x) = xT_(α⁻)^(α₊)(x),

where

T_(α⁻)^(α₊)(x)

is a transparency function with the characteristic transparency range[α_, α₊]. We may require that

T_(α⁻)^(α₊)(x)

is effectively (or approximately) unity for α_ ≤ x ≤ a₊, and that

T_(α⁻)^(α₊)(|x|)

becomes smaller than unity (e.g. decays to zero) for x outside of therange [α_, α₊].

As one should be able to see in FIG. 3 , a (nonlinear) differentialequation relating the input x(t) to the output χ(t) of an ABAINF may bewritten as

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\chi = \frac{1}{\tau}I_{\alpha_{-}}^{\alpha_{+}}\left( {x - \chi} \right) = \frac{x - \chi}{\tau}T_{\alpha_{-}}^{\alpha_{+}}\left( {x - \chi} \right),} & \text{­­­(12)}\end{matrix}$

where τ is the ABAINF’s time parameter (or time constant).

One skilled in the art will recognize that, according to equation (12),when the difference signal x(t) - χ(t) is within the transparency range[α₋, α₊], the ABAINF would behave as a 1st order linear lowpass filterwith the 3 dB corner frequency 1/(2πτ), and, for a sufficiently largetransparency range, the ABAINF would exhibit nonlinear behavior onlyintermittently, when the difference signal extends outside thetransparency range.

If the transparency range [α₋, α₊] is chosen in such a way that itexcludes outliers of the difference signal x(t) - χ(t), then, since thetransparency function

T_(α⁻)^(α₊)(x)

decreases (e.g. decays to zero) for x outside of the range [α₋, α₊], thecontribution of such outliers to the output χ(t) would be depreciated.

It may be important to note that outliers would be depreciateddifferentially, that is, based on the difference signal x(t) - χ(t) andnot the input signal x(t).

The degree of depreciation of outliers based on their magnitude woulddepend on how rapidly the transparency function

T_(α⁻)^(α₊)(x)

decreases (e.g. decays to zero) for x outside of the transparency range.For example, as follows from equation (12), once the transparencyfunction decays to zero, the output χ(t) would maintain a constant valueuntil the differencesignal x(t) - χ(t) returns to within non-zero valuesof the transparency function.

FIG. 4 provides several illustrative examples of the transparencyfunctions and their respective influence functions.

Note that panel (viii) in FIG. 4 provides an example of unboundedinfluence function, when the respective transparency function may notdecay to zero,

$\begin{matrix}{I_{\alpha} = xT_{\alpha}(x) = x \times \left\{ \begin{matrix}1 & {\text{­­­(13)}|x| \leq \alpha} \\\varepsilon & \text{otherwise}\end{matrix} \right)\mspace{6mu}\mspace{6mu},} & \end{matrix}$

where ε ≤ 0. Also note that for the particular influence function shownin panel (viii) of FIG. 4 the ABAINF’s behavior outside the transparencyrange will be linear, albeit different from the behavior when thedifference signal x(t) - χ(t) is within the transparency range [α₋, a₊].

One skilled in the art will recognize that a transparency function withmultiple transparency ranges may also be constructed as a product of(e.g. cascaded) transparency functions, wherein each transparencyfunction is characterized by its respective transparency range.

2.1 A Particular ABAINF Example

As an example, let us consider a particular ABAINF with the influencefunction of a type shown in panel (iii) of FIG. 4 , for a symmetricaltransparency range [α₋, α₊] = [-α, α]:

$\begin{matrix}{I_{\alpha} = xT_{\alpha}(x) = x \times \left\{ \begin{matrix}1 & {\text{­­­(14)}|x| \leq \alpha} \\\frac{\mu\tau}{|x|} & \text{otherwise}\end{matrix} \right)\mspace{6mu}\mspace{6mu},} & \end{matrix}$

where α ≥ 0 is the resolution parameter (with units “amplitude”), τ ≥ 0is the time parameter (with units “time”), and µ ≥ 0 is the rateparameter (with units “amplitude per time”).

For such an ABAINF, the relation between the input signal x(t) and thefiltered output signal χ(t) may be expressed as

$\begin{matrix}{\overset{˙}{\chi} = \frac{x - \chi}{\tau}\left\lbrack {\theta\left( {\alpha - \left| {x - \chi} \right|} \right) + \frac{\mu\tau}{\left| {x - \chi} \right|}\theta\left( {\left| {x - \chi} \right| - \alpha} \right)} \right\rbrack,} & \text{­­­(15)}\end{matrix}$

where θ(x) is the Heaviside unit step function [30].

Note that when |x - χ| ≤ α (e.g., in the limit α → ∞) equation (15)describes a 1st order analog linear lowpass filter (RC integrator) withthe time constant τ (the 3 dB corner frequency 1/(2πτ)). When themagnitude of the difference signal |x-χ| exceeds the resolutionparameter α, however, the rate of change of the output would be limitedto the rate parameter µ and would no longer depend on the magnitude ofthe incoming signal x(t), providing a robust output (i.e. an outputinsensitive to outliers with a characteristic amplitude determined bythe resolution parameter α). Note that for a sufficiently large α thisfilter would exhibit nonlinear behavior only intermittently, in responseto noise outliers, while otherwise acting as a 1st order linear lowpassfilter.

Further note that for µ = α/τ equation (15) corresponds to the CanonicalDifferential Limiter (CDL) described in [9, 10, 24, 32], and in thelimit α → 0 it corresponds to the Median Tracking Filter described in§3.1.

However, an important distinction of this ABAINF from the nonlinearfilters disclosed in [9, 10, 24, 32] would be that the resolution andthe rate parameters are independent from each other. This may providesignificant benefits in performance, ease of implementation, costreduction, and in other areas, including those clarified and illustratedfurther in this disclosure.

2.2 Clipped Mean Tracking Filter (CMTF)

The blanking influence function shown in FIG. 4(i) would be anotherparticular example of the ABAINF outlined in FIG. 3 , where thetransparency function may be represented as a boxcar function,

$\begin{matrix}{T_{\alpha_{-}}^{\alpha_{+}}(x) = \theta\left( {x - \alpha_{-}} \right) - \theta\left( {x - \alpha_{+}} \right).} & \text{­­­(16)}\end{matrix}$

For this particular choice, the ABAINF may be represented by thefollowing 1st order nonlinear differential equation:

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\chi = \frac{1}{\tau}B_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}\left( {x - \chi} \right),} & \text{­­­(17)}\end{matrix}$

where the blanking function

B_( α⁻)^(α₊)(x)

may be defined as

$\begin{matrix}{B_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}(x) = \left\{ \begin{matrix}x & {\text{­­­(18)}\alpha_{-} \leq x \leq \alpha_{+}} \\0 & \text{otherwise}\end{matrix} \right)\mspace{6mu},} & \end{matrix}$

and where [α₋, α₊] may be called the blanking range.

We shall call an ABAINF with such influence function a 1st order ClippedMean Tracking Filter (CMTF).

A block diagram of a CMTF is shown in FIG. 5 (a) . In this figure, theblanker implements the blanking function

B_( α⁻)^(α₊)(x).

In a similar fashion, we may call a circuit implementing an influencefunction

I_(α⁻)^(α₊)(x)

a depreciator with characteristic depreciation (or transparency, orinfluence) range [α₋, α₊].

Note that, for b > 0,

$\begin{matrix}{b^{- 1}I_{\alpha_{-}}^{\alpha_{+}}\left( {bx} \right) = I_{\frac{\alpha_{-}}{b}}^{\frac{\alpha_{+}}{b}}(x),} & \text{­­­(19)}\end{matrix}$

and thus, if the blanker with the range [V₋, V₊] is preceded by a gainstage with the gain G and followed by a gain stage with the gain G⁻¹,its apparent (or “equivalent”) blanking range would be [V₋, V₊]/G, andwould no longer be hardware limited. Thus control of transparency rangesof practical ABAINF implementations may be performed by automatic gaincontrol (AGC) means. This may significantly simplify practicalimplementations of ABAINF circuits (e.g. by allowing constant hardwaresettings for the transparency ranges). This is illustrated in FIG. 5 (b)for the CMTF circuit.

FIG. 6 illustrates resistance of a CMTF (with a symmetrical blankingrange [-α, α]) to outlier noise, in comparison with a 1st order linearlowpass filter with the same time constant (panel (a)), and with the CDLwith the resolution parameter α and τ₀ = τ (panel (b)). Thecross-hatched time intervals in the lower panel (panel (c)) correspondto nonlinear CMTF behavior (zero rate of change of the output). Notethat the clipping (i.e. zero rate of change of the CMTF output) isperformed differentially, based on the magnitude of the differencesignal x(t) - χ(t) and not that of the input signal x(t).

We may call the difference between a filter output when the input signalis affected by impulsive noise and an “ideal” output (in the absence ofimpulsive noise) an “error signal”. Then the smaller the error signal,the better the impulsive noise suppression. FIG. 7 illustratesdifferences in the error signal for the example of FIG. 6 . Thecross-hatched time intervals indicate nonlinear CMTF behavior (zero rateof change).

2.3 Illustrative CMTF Circuit

FIG. 8 provides a simplified illustration of implementing a CMTF bysolving equation (17) in an electronic circuit.

FIG. 9 provides an illustration of resistance of the CMTF circuit ofFIG. 8 to outlier noise. The cross-hatched time intervals in the lowerpanel correspond to nonlinear CMTF behavior.

While FIG. 8 illustrates implementation of a CMTF in an electroniccircuit comprising discrete components, one skilled in the art willrecognize that the intended electronic functionality may be implementedby discrete components mounted on a printed circuit board, or by acombination of integrated circuits, or by an application-specificintegrated circuit (ASIC). Further, one skilled in the art willrecognize that a variety of alternative circuit topologies may bedeveloped and/or used to implement the intended electronicfunctionality.

2.4 Using CMTFs for Separating Impulsive (Outlier) and Non-ImpulsiveSignal Components With Overlapping Frequency Spectra: AnalogDifferential Clippers (ADiCs)

In some applications it may be desirable to separate impulsive (outlier)and non-impulsive signal components with overlapping frequency spectrain time domain.

Examples of such applications would include radiation detectionapplications, and/or dual function systems (e.g. using radar as signalof opportunity for wireless communications and/or vice versa).

Such separation may be achieved by using sums and/or differences of theinput and the output of a CMTF and its various intermediate signals.This is illustrated in FIG. 10 .

In this figure, the difference between the input to the CMTF integrator(signal τ χ̇(t) at point III) and the CMTF output may be designated as aprime output of an Analog Differential Clipper (ADiC) and may beconsidered to be a non-impulsive (“background”) component extracted fromthe input signal. Further, the signal across the blanker (i.e. thedifference between the blanker input x(t) - χ(t) and the blanker outputτ χ̇(t)) may be designated as an auxiliary output of an ADiC and may beconsidered to be an impulsive (outlier) component extracted from theinput signal.

FIG. 11 illustrates using a CMTF with an appropriately chosen blankingrange for separating impulsive and non-impulsive (“background”) signalcomponents. Note that the sum of the prime and the auxiliary ADiCoutputs would be effectively identical to the input signal, and thus theseparation of impulsive and non-impulsive components may be achievedwithout reducing signal’s bandwidth.

FIG. 12 provides illustrative block diagrams of an ADiC with timeparameter τ and blanking range [α₋, a₊]. In the figure, x(t) is the ADiCinput, and y(t) is the (“prime”) ADiC output. We may call the“intermediate” signal χ(t) (the CMTF output) the Differential ClippingLevel, and the blanker input is the “difference signal” x(t) - χ(t). Theblanker output equals to its input if it falls within the blanking range[α₋, α₊]. Otherwise, this output is zero.

For a robust (i.e insensitive to outliers) blanking range [α₋, α₊]around the difference signal, the portion of the difference signal thatprotrudes from this range may be identified as an outlier. As may beseen in FIG. 12 , when the blanker’s output is zero (that is, an outlieris encountered), χ(t) would be maintained at its previous level. As theresult, in the ADiC’s output the outliers would be replaced by theDifferential Clipping Level χ(t), otherwise the signal would not beaffected.

FIG. 13 provides a simplified illustrative electronic circuit diagram ofusing a CMTF/ADiC with an appropriately chosen blanking range [α₋, α₊]for separating incoming signal x(t) into impulsive i(t) andnon-impulsive s(t) (“background”) signal components, and FIG. 14provides an illustration of such separation by the circuit of FIG. 13 .

Note that while a blanker used in the ADiC shown in FIG. 12 , adepreciator described by a different transparency function (e.g. one ofthose shown in FIG. 4 ) may be used. In such a case, the ADiC output maybe given by the following equation:

$\begin{matrix}{\left\{ \begin{matrix}{y(t) = \chi(t) + \tau\overset{˙}{\chi}(t)} \\{\overset{˙}{\chi}(t) = \frac{1}{\tau}T_{\alpha_{-}}^{\alpha_{+}}\left( {x(t) - \chi(t)} \right)}\end{matrix} \right)\mspace{6mu}\mspace{6mu}.} & \text{­­­(20)}\end{matrix}$

As may be seen from equation (20), when the difference signal x(t) -χ(t) is within the transparency range [α₋, α₊], then the ADiC outputy(t) equals to its input x(t) (y(t) = χ(t)+[x(t) - χ(t)] = x(t)).However, when the difference signal is outside the transparency range(i.e an outlier is detected), the value of the transparency function issmaller then zero (for example, it is ε < 1) and thus

T_(α⁻)^(α₊)(x(t) − X)^(((t))) = ε[x(t) − x(t)]

and the outlier is depreciated (e.g. in the ADiC output the outlier isreplaced by y(t) = x(t) + ε [x(t) - χ(t)]).

2.5 Numerical Implementations of ABAINFs/CMTFs/ADiCs

Even though an ABAINF is an analog filter by definition, it may beeasily implemented digitally, for example, in a Field Programmable GateArray (FPGA) or software. A digital ABAINF would require very littlememory and would be typically inexpensive computationally, which wouldmake it suitable for real-time implementations.

An example of a numerical algorithm implementing a finite-differenceversion of a CMTF/ADiC may be given by the following MATLAB function:

function [chi,prime,aux] = CMTF_ADiC(x,t,tau,alpha_p,alpha_m)  chi = zeros(size(x));   aux = zeros(size(x));  prime = zeros(size(x));   dt = diff (t) ;   chi(1) = x(1);   B = 0;  for i = 2 : length (x) ;     dX = x ( i ) - chi(i-1);    if dX>alpha_p(i-1)       B = 0;     elseif dX<alpha_m(i-1)      B = 0;     else       B = dX;     end    chi ( i ) = chi(i-1) + B/(tau+dt(i-1))*dt(i-1); % numerical antiderivative    prime(i) = B + chi(i-1);     aux(i) = dX - B;   end return

In this example, “x” is the input signal, “t” is the time array, “tau”is the CMTF’s time constant, “alpha_p” and “alpha_m” are the upper andthe lower, respectively, blanking values, “chi” is the CMTF’s output,“aux” is the extracted impulsive component (auxiliary ADiC output), and“prime” is the extracted non-impulsive (“background”) component (primeADiC output).

Note that we retain, for convenience, the abbreviations “ABAINF” and/or“ADiC” for finite-difference (digital) ABAINF and/or ADiCimplementations.

FIG. 15 provides an illustration of separation of a discrete inputsignal “x” into an impulsive component “aux” and a non-impulsive(“background”) component “prime” using the above MATLAB function withappropriately chosen blanking values “alpha_p” and “alpha_m”.

A digital signal processing apparatus performing an ABAINF filteringfunction transforming an input signal into an output filtered signalwould comprise an influence function characterized by a transparencyrange and operable to receive an influence function input and to producean influence function output, and an integrator function characterizedby an integration time constant and operable to receive an integratorinput and to produce an integrator output, wherein said integratoroutput is proportional to a numerical antiderivative of said integratorinput.

A hardware implementation of a digital ABAINF/CMTF/ADiC filteringfunction may be achieved by various means including, but not limited to,general-purpose and specialized microprocessors (DSPs),microcontrollers, FPGAs, ASICs, and ASSPs. A digital or a mixed-signalprocessing unit performing such a filtering function may also perform avariety of other similar and/or different functions.

3 Quantile Tracking Filters as Robust Means to Establish the ABAINFTransparency Range(s)

Let y(t) be a quasi-stationary signal with a finite interquartile range(IQR), characterized by an average crossing rate 〈f〉 of the thresholdequal to some quantile q, 0 < q < 1, of y(t). (See [33, 34] fordiscussion of quantiles of continuous signals, and [46, 47] fordiscussion of threshold crossing rates.) Let us further consider thesignal Q_(q)(t) related to y(t) by the following differential equation:

$\begin{matrix}{\frac{\text{d}}{\text{d}t}Q_{q} = \frac{A}{T}\left\lbrack {{sgn}\left( {y - Q_{q}} \right) + 2q - 1} \right\rbrack\mspace{6mu},} & \text{­­­(21)}\end{matrix}$

where A is a parameter with the same units as y and Q_(q), and T is aconstant with the units of time. According to equation (21), Q_(q)(t) isa piecewise-linear signal consisting of alternating segments withpositive (2qA/T) and negative (2(q - 1)A/T) slopes. Note that Q_(q)(t) ≈const for a sufficiently small A/T (e.g., much smaller than the productof the IQR and the average crossing rate 〈f〉 of y(t) and its q thquantile), and a steady-state solution of equation (21) can be writtenimplicitly as

$\begin{matrix}{\overline{\theta\left( {Q_{q}-y} \right)} \approx q,} & \text{­­­(22)}\end{matrix}$

where θ(x) is the Heaviside unit step function [30] and the overlinedenotes averaging over some time interval ΔT >> 〈f〉⁻¹. Thus Q_(q)would approximate the qth quantile of y(t) [33, 34] in the time intervalΔT.

We may call an apparatus (e.g. an electronic circuit) effectivelyimplementing equation (21) a Quantile Tracking Filter.

Despite its simplicity, a circuit implementing equation (21) may providerobust means to establish the ABAINF transparency range(s) as a linearcombination of various quantiles of the difference signal (e.g. its 1stand 3rd quartiles and/or the median). We will call such a circuit for q= ½ a Median Tracking Filter (MTF), and for q = ¼ and/or q = ¾ - aQuartile Tracking Filter (QTF).

FIG. 16 provides an illustrative block diagram of a circuit implementingequation (21) and thus tracking a q th quantile of y(t). As one may seein the figure, the difference between the input y(t) and the quantileoutput Q_(q)(t) forms the input to an analog comparator which implementsthe function A sgn (y(t)-Q_(q)(t)). In reference to FIG. 16 , we maycall the term (2q - 1)A added to the integrator input as the “quantilesetting signal”. A sum of the comparator output and the quantile settingsignal forms the input of an integrator characterized by the timeconstant T, and the output of the integrator forms the quantile outputQ_(q)(t).

3.1 Median Tracking Filter

Let x(t) be a quasi-stationary signal characterized by an averagecrossing rate 〈f〉 of the threshold equal to the second quartile(median) of x(t). Let us further consider the signal Q₂(t) related tox(t) by the following differential equation:

$\begin{matrix}{\frac{\text{d}}{\text{d}t}Q_{2} = \frac{A}{T}{sgn}\left( {x - Q_{2}} \right),} & \text{­­­(23)}\end{matrix}$

where A is a constant with the same units as x and Q₂, and T is aconstant with the units of time. According to equation (23), Q₂(t) is apiecewise-linear signal consisting of alternating segments with positive(A/T) and negative (-A/T) slopes. Note that Q₂(t) ≈ const for asufficiently small A/T (e.g., much smaller than the product of theinterquartile range and the average crossing rate 〈f〉 of x(t) and itssecond quartile), and a steady-state solution of equation (23) may bewritten implicitly as

$\begin{matrix}{\overline{\theta\left( {Q_{2}-x} \right)} \approx \frac{1}{2},} & \text{­­­(24)}\end{matrix}$

where the overline denotes averaging over some time interval ΔT >>〈f〉⁻¹. Thus Q₂ approximates the second quartile of x(t) in the timeinterval ΔT, and equation (23) describes a Median Tracking Filter (MTF).FIG. 17 illustrates the MTF’s convergence to the steady state fordifferent initial conditions.

3.2 Quartile Tracking Filters

Let y(t) be a quasi-stationary signal with a finite interquartile range(IQR), characterized by an average crossing rate 〈f〉 of the thresholdequal to the third quartile of y(t). Let us further consider the signalQ₃(t) related to y(t) by the following differential equation:

$\begin{matrix}{\frac{\text{d}}{\text{d}t}Q_{3} = \frac{A}{T}\left\lbrack {{sgn}\left( {y - Q_{3}} \right) + \frac{1}{2}} \right\rbrack,} & \text{­­­(25)}\end{matrix}$

where A is a constant (with the same units as y and Q₃), and T is aconstant with the units of time. According to equation (25), Q₃(t) is apiecewise-linear signal consisting of alternating segments with positive(3A/(2T)) and negative (-A/(2T)) slopes. Note that Q₃(t) ≈ const for asufficiently small A/T (e.g., much smaller than the product of the IQRand the average crossing rate 〈f〉 of y(t) and its third quartile), anda steady-state solution of equation (25) may be written implicitly as

$\begin{matrix}{\overline{\theta\left( {Q_{3}-y} \right)} \approx \frac{3}{4},} & \text{­­­(26)}\end{matrix}$

where the overline denotes averaging over some time interval ΔT >>〈f〉⁻¹. Thus Q₃ approximates the third quartile of y(t) [33, 34] in thetime interval ΔT.

Similarly, for

$\begin{matrix}{\frac{\text{d}}{\text{d}t}Q_{1} = \frac{A}{T}\left\lbrack {{sgn}\left( {y - Q_{1}} \right) - \frac{1}{2}} \right\rbrack} & \text{­­­(27)}\end{matrix}$

a steady-state solution may be written as

$\begin{matrix}{\overline{\theta\left( {Q_{1}-y} \right)} \approx \frac{1}{4},} & \text{­­­(28)}\end{matrix}$

and thus Q₁ would approximate the first quartile of y(t) in the timeinterval ΔT.

FIG. 18 illustrates the QTFs′ convergence to the steady state fordifferent initial conditions.

One skilled in the art will recognize that (1) similar tracking filtersmay be constructed for other quantiles (such as, for example, terciles,quintiles, sextiles, and so on), and (2) a robust range [α₋, α₊] thatexcludes outliers may be constructed in various ways, as, for example, alinear combination of various quantiles.

3.3 Numerical Implementations of ABAINFs/CMTFs/ADiCs Using QuantileTracking Filters as Robust Means to Establish the Transparency Range

For example, an ABAINF/CMTF/ADiC with an adaptive (possibly asymmetric)transparency range [α₋, α₊] may be designed as follows. To ensure thatthe values of the difference signal x(t) - χ(t) that lie outside of [α₋,α₊] are outliers, one may identify [α₋, a₊] with Tukey’s range [48], alinear combination of the 1st (Q₁) and the 3rd (Q₃) quartiles of thedifference signal:

$\begin{matrix}{\left\lbrack {\alpha_{-},\alpha_{+}} \right\rbrack = \left\lbrack {Q_{1} - \beta\left( {Q_{3} - Q_{1}} \right),Q_{3} + \beta\left( {Q_{3} - Q_{1}} \right)} \right\rbrack,} & \text{­­­(29)}\end{matrix}$

where β is a coefficient of order unity (e.g. β = 1.5).

An example of a numerical algorithm implementing a finite-differenceversion of a CMTF/ADiC with the blanking range computed as Tukey’s rangeof the difference signal using digital QTFs may be given by the MATLABfunction “CMTF_ADiC_alpha” below.

In this example, the CMTF/ADiC filtering function further comprises ameans of tracking the range of the difference signal that effectivelyexcludes outliers of the difference signal, and wherein said meanscomprises a QTF estimating a quartile of the difference signal:

function [chi,prime,aux,alpha_p,alpha_m] = CMTF_ADiC_alpha(x,t,tau,beta,mu)  chi = zeros(size(x));   aux = zeros(size(x));  prime = zeros(size(x));   alpha_p = zeros(size(x));  alpha_m = zeros(size(x));   dt = diff (t) ;   chi(1) = x(1);  Q1 = x ( 1 ) ;   Q3 = x (1) ;   B = 0;   for i = 2 : length (x) ;    dX = x (i) - chi(i-1);  %--------------------------------------------------------------------  % Update 1st and 3rd quartile values:    Q1 = Q1 + mu_(*)(sign(dX-Q1)-0.5)*dt(i-1); % numerical antiderivative    Q3 = Q3 + mu_(*)(sign(dX-Q3)+0.5)*dt(i-1); % numerical antiderivative  %--------------------------------------------------------------------  % Calculate blanking range:     alpha_p(i) = Q3 + beta_(*)(Q3-Q1);    alpha_m(i) = Q1 - beta_(*)(Q3-Q1);  %--------------------------------------------------------------------    if dX>alpha_p(i)       B = 0;     elseif dX<alpha_m(i)       B = 0;    else       B = dX;     end    chi(i) = chi(i-1) + B/(tau+dt(i-1))*dt(i-1); % numerical antiderivative    prime(i) = B + chi(i-1);     aux(i) = dX - B;   end return

FIG. 19 provides an illustration of separation of discrete input signal“x” into impulsive component “aux” and non-impulsive (“background”)component “prime” using the above MATLAB function of §3.3 with theblanking range computed as Tukey’s range using digital QTFs. The upperand lower limits of the blanking range are shown by the dashed lines inpanel (b).

Since outputs of analog QTFs are piecewise-linear signals consisting ofalternating segments with positive and negative slopes, a care should betaken in finite difference implementations of QTFs wheny(n)-Q_(q)(n - 1) is outside of the interval hA[2(q - 1), 2q] /T, whereh is the time step. For example, in such a case one may set Q_(q)(n) =y(n), as illustrated in the example below.

function [xADiC,xCMTF,resid,alpha_p,alpha_m]=ADiC_IQRscaling(x,dt,tau,beta,mu)  Ntau = (1+floor(tau/dt));  xADiC = zeros(size(x)); xCMTF = zeros(size(x)); resid = zeros(size(x));  alpha_p = zeros(size(x)); alpha_m = zeros(size(x)); gamma = mu∗dt;  xADiC(1) = x(1); xCMTF(1) = x(1); Balphapm = 0; Q1 = x(1); Q3 = x(1);  for i = 2:length(x);     dX = x(i)-xCMTF(i-1);  %--------------------------------------------------------------------  % Update 1st and 3rd quartile values:     dX3 = dX - Q3;    if dX3 > -gamma/2 & dX3 < 3∗gamma/2         Q3 = dX;     else        Q3 = Q3 + gamma∗(sign(dX3)+0.5);     end     dX1 = dX - Q1;    if dX1 > -3∗gamma/2 & dX1 < gamma/2         Q1 = dX;     else        Q1 = Q1 + gamma∗(sign(dX1)-0.5);     end  %--------------------------------------------------------------------  % Calculate blanking range:    alpha_p(i) = Q3 + beta∗(Q3-Q1); alpha_m(i) = Q1 - beta∗(Q3-Q1);  %--------------------------------------------------------------------    M = (Q3+Q1)/2; R = (1+2∗beta)∗(Q3-Q1)/2;     if dX>alpha_p(i)+1e-12      Balphapm = dX∗(R/(dX-M))^2;     elseif dX<alpha_m(i)-1e-12      Balphapm = dX∗(R/(dX-M))^2;     else       Balphapm = dX;     end    xCMTF(i) = xCMTF(i-1) + Balphapm/Ntau;    xADiC(i) = Balphapm + xCMTF(i-1); resid(i) = dX - Balphapm; endreturn

Note that in this example the following transparency function is used:

$\begin{matrix}{T_{\alpha_{-}}^{\alpha_{+}}(x) = \left\{ \begin{array}{l}{1\quad\text{for}\quad a_{-} \leq x \leq \alpha +} \\{\left( \frac{R}{x - M} \right)^{2}\quad\text{otherwise}}\end{array} \right),} & \text{­­­(30)}\end{matrix}$

where

$R = \frac{\alpha_{+} - \alpha_{-}}{2}\mspace{6mu}\text{and}\mspace{6mu} M = \frac{\alpha_{+} + \alpha_{-}}{2}.$

This transparency function is illustrated in FIG. 20 .

3.4 Adaptive Influence Function Design

The influence function choice determines the structure of the localnonlinearity imposed on the input signal. If the distribution of thenon-Gaussian technogenic noise is known, then one may invoke the classiclocally most powerful (LMP) test [49] to detect and mitigate the noise.The LMP test involves the use of local nonlinearity whose optimal choicecorresponds to

$g_{lo}(n) = - \frac{f^{\prime}(n)}{f(n)},$

where f (n) represents the technogenic noise density function and f′ (n)is its derivative. While the LMP test and the local nonlinearity istypically applied in the discrete time domain, the present inventionenables the use of this idea to guide the design of influence functionsin the analog domain. Additionally, non-stationarity in the noisedistribution may motivate an online adaptive strategy to designinfluence functions.

Such adaptive online influence function design strategy may explore themethodology disclosed herein. In order to estimate the influencefunction, one may need to estimate both the density and its derivativeof the noise. Since the difference signal x(t)-χ(t) of an ABAINF wouldeffectively represent the non-Gaussian noise affecting the signal ofinterest, one may use a bank of N quantile tracking filters described in§3 to determine the sample quantiles (Q₁, Q₂, ..., Q_(N)) of thedifference signal. Then one may use a non-parametric regressiontechnique such as, for example, a local polynomial kernel regressionstrategy to simultaneously estimate (1) the time-dependent amplitudedistribution function Φ(D, t) of the difference signal, (2) its densityfunction ϕ(D, t), and (3) the derivative of the density function ∂ϕ(D,t)/∂D.

4 Adaptive Intermittently Nonlinear Analog Filters for Mitigation ofOutlier Noise in the Process of Analog-to-Digital Conversion

Let us now illustrate analog-domain mitigation of outlier noise in theprocess of analog-to-digital (A/D) conversion that may be performed bydeploying an ABAINF (for example, a CMTF) ahead of an ADC.

An illustrative principal block diagram of an adaptive CMTF formitigation of outlier noise disclosed herein is shown in FIG. 21 .Without loss of generality, here it may be assumed that the outputranges of the active components (e.g. the active filters, integrators,and comparators), as well as the input range of the analog-to-digitalconverter (A/D), are limited to a certain finite range, e.g., to thepower supply range ±V_(c).

The time constant τ may be such that 1/(2πτ) is similar to the cornerfrequency of the anti-aliasing filter (e.g., approximately twice thebandwidth of the signal of interest B_(x)), and the time constant Tshould be two to three orders of magnitude larger than

B_(x)⁻¹.

The purpose of the front-end lowpass filter would be to sufficientlylimit the input noise power. However, its bandwidth may remainsufficiently wide (i.e. γ >> 1) so that the impulsive noise is notexcessively broadened.

Without loss of generality, we may further assume that the gain K isconstant (and is largely determined by the value of the parameter γ,e.g., as

$K \sim \sqrt{\text{γ}}$

), and the gains G and g are adjusted (e.g. using automatic gaincontrol) in order to well utilize the available output ranges of theactive components, and the input range of the A/D. For example, G and gmay be chosen to ensure that the average absolute value of the outputsignal (i.e., observed at point IV) is approximately V_(c)/5, and theaverage value of

Q₂^(⋆)(t)

is approximately constant and is smaller than V_(c).

4.1 CMTF Block

For the Clipped Mean Tracking Filter (CMTF) block shown in FIG. 21 , theinput x(t) and the output χ(t) signals may be related by the following1st order nonlinear differential equation:

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\chi = \frac{1}{\tau} = B_{\frac{V_{c}}{g}}\left( {x - \chi} \right),} & \text{­­­(31)}\end{matrix}$

where the symmetrical blanking function B_(α)(x) may be defined as

$\begin{matrix}{B_{\alpha}(x) = \left\{ \begin{array}{l}{x\quad\text{for}\quad|x| \leq \alpha} \\{0\quad\text{otherwise}}\end{array} \right),} & \text{­­­(32)}\end{matrix}$

and where the parameter α is the blanking value.

Note that for the blanking values such that |x(t)-χ(t)| ≤ V_(c/)g forall t, equation (31) describes a 1st order linear lowpass filter withthe corner frequency 1/(2πτ), and the filter shown in FIG. 21 operatesin a linear regime (see FIG. 22 ). However, when the values of thedifference signal x(t)-χ(t) are outside of the interval [-V_(c)/g,V_(c)/g], the rate of change of χ(t) is zero and no longer depends onthe magnitude of x(t)-χ(t). Thus, if the values of the difference signalthat lie outside of the interval [-V_(c)/g, V_(c)/g] are outliers, theoutput χ(t) would be insensitive to further increase in the amplitude ofsuch outliers. FIG. 6 illustrates resistance of a CMTF to outlier noise,in comparison with a 1st order linear lowpass filter with the same timeconstant. The shaded time intervals correspond to nonlinear CMTFbehavior (zero rate of change). Note that the clipping (i.e. zero rateof change of the CMTF output) is performed differentially, based on themagnitude of the difference signal |x - χ| and not that of the inputsignal x.

In the filter shown in FIG. 21 the range [-V_(c)/g, V_(c)/g] thatexcludes outliers is obtained as Tukey’s range [48] for a symmetricaldistribution, with V_(c)/g given by

$\begin{matrix}{\frac{V_{c}}{g} = \left( {1 + 2\beta} \right)Q_{2}^{\star},} & \text{­­­(33)}\end{matrix}$

where

Q₂^(⋆)

is the 2nd quartile (median) of the absolute value of the differencesignal |x(t)-χ(t)|, and where β is a coefficient of order unity (e.g. β= 3). While in this example we use Tukey’s range, various alternativeapproaches to establishing a robust interval [-V_(c)/g, V_(c)/g] may beemployed.

In FIG. 21 , the MTF circuit receiving the absolute value of the blankerinput and producing the MTF output, together with a means to maintainsaid MTF output at approximately constant value (e.g. using automaticgain control) and the gain stages preceding and following the blanker,establish a blanking range that effectively excludes outliers of theblanker input.

It would be important to note that, as illustrated in panel I of FIG. 22, in the linear regime the CMTF would operate as a 1st order linearlowpass filter with the corner frequency 1/(2πτ). It would exhibitnonlinear behavior only intermittently, in response to outliers in thedifference signal, thus avoiding the detrimental effects, such asinstabilities and intermodulation distortions, often associated withnonlinear filtering.

4.2 Baseband Filter

In the absence of the CMTF in the signal processing chain, the basebandfilter following the A/D would have the impulse response w[k] that maybe viewed as a digitally sampled continuous-time impulse response w(t)(see panel II of FIG. 22 ). As one may see in FIG. 21 , the impulseresponse of this filter may be modified by adding the term τẇ[k], wherethe dot over the variable denotes its time derivative, and where ẇ[k]may be viewed as a digitally sampled continuous-time function ẇ(t). Thisadded term would compensate for the insertion of a 1st order linearlowpass filter in the signal chain, as illustrated in FIG. 22 .

Indeed, from the differential equation for a 1st order lowpass filter itwould follow that h_(τ) ∗ (w + τẇ) = w, where the asterisk denotesconvolution and where h_(τ)(t) is the impulse response of the 1st orderlinear lowpass filter with the corner frequency 1/(2πτ). Thus, providedthat τ is sufficiently small (e.g., τ ≲ 1/(2πB_(aa)), where B_(aa) isthe nominal bandwidth of the anti-aliasing filter), the signal chainsshown in panels I and II of FIG. 22 would be effectively equivalent. Theimpulse and frequency responses of w[k] (a root-raised-cosine filterwith the roll-off factor ¼, bandwidth 5B_(x)/4, and the sampling rate8B_(x)) and w[k] + τẇ[k] (with τ = 1/(4πB_(x))) used in the subsequentexamples of this section are shown in FIG. 23 .

4.3 Comparative Performance Examples 4.3.1 Simulation Parameters

To emulate the analog signals in the simulated examples presented below,the digitisa-tion rate was chosen to be significantly higher (by abouttwo orders of magnitude) than the A/D sampling rate.

The signal of interest is a Gaussian baseband signal in the nominalfrequency rage [0, B_(x)]. It is generated as a broadband white Gaussiannoise filtered with a root-raised-cosine filter with the roll-off factor¼ and the bandwidth 5B_(x)/4.

The noise affecting the signal of interest is a sum of an Additive WhiteGaussian Noise (AWGN) background component and white impulsive noisei(t). In order to demonstrate the applicability of the proposed approachto establishing a robust interval [-V_(c)/g, V_(c)/g] for asymmetricaldistributions, the impulsive noise is modelled as asymmetrical(unipolar) Poisson shot noise:

$\begin{matrix}{i(t) = \left| {v(t)} \right|{\sum\limits_{k = 1}^{\infty}{\delta\left( {t - t_{k}} \right)}},} & \text{­­­(34)}\end{matrix}$

where v(t) is AWGN noise, t_(k) is the k-th arrival time of a Poissonpoint process with the rate parameter λ, and δ(x) is the Diracδ-function [31]. In the examples below, λ = 2B_(x).

The A/D sampling rate is 8B, (that assumes a factor of 4 oversampling ofthe signal of interest), the A/D resolution is 12 bits, and theanti-aliasing filter is a 2nd order Butterworth lowpass filter with thecorner frequency 2B_(x). Further, the range of the comparators in theQTFs is ±A = ±V_(c), the time constants of the integrators are τ =1/(4πB_(x)) and T = 100/B_(x). The impulse responses of the basebandfilters w[k] and w[k] + τẇ[k] are shown in the upper panel of FIG. 23 .

The front-end lowpass filter is a 2nd order Bessel with the cutofffrequency γ/(2πτ). The value of the parameter γ is chosen as γ = 16, andthe gain of the anti-aliasing filter is

$K = \sqrt{\text{γ}} = 4.$

The gains G and g are chosen to ensure that the average absolute valueof the output signal (i.e., observed at point IV in FIG. 21 and at point(c) in FIG. 22 ) is approximately V_(c)/5, and

$Q_{2}^{\star}(t) \approx \frac{V_{c}}{1 + 2\beta} = \text{const}.$

4.3.2 Comparative Channel Capacities

For the simulation parameters described above, FIG. 24 compares thesimulated channel capacities (calculated from the baseband SNRs usingthe Shannon formula [44]) for various signal+noise compositions, for thelinear signal processing chain shown in panel II of FIG. 22 (solidcurves) and the CMTF-based chain of FIG. 21 with β = 3 (dotted curves).

As one may see in FIG. 24 (and compare with the simplified diagram ofFIG. 2 ), for a sufficiently large β both linear and the CMTF-basedchains provide effectively equivalent performance when the AWGNdominates over the impulsive noise. However, the CMTF-based chains areinsensitive to further increase in the impulsive noise when the latterbecomes comparable or dominates over the thermal (Gaussian) noise, thusproviding resistance to impulsive interference.

Further, the dashed curves in FIG. 24 show the simulated channelcapacities for the CMTF-based chain of FIG. 21 (with β = 3) whenadditional interference in an adjacent channel is added, as would be areasonably common practical scenario. The passband of this interferenceis approximately [3B_(x), 4B_(x)], and the total power is approximately4 times (6 dB) larger that that of the signal of interest. As one maysee in FIG. 24 , such interference increases the apparent blanking valueneeded to maintain effectively linear CMTF behaviour in the absence ofthe outliers, reducing the effectiveness of the impulsive noisesuppression (more noticeably for higher AWGN SNRs).

It may be instructive to illustrate and compare the changes in thesignal’s time and frequency domain properties, and in its amplitudedistributions, while it propagates through the signal processing chains,linear (points (a), (b), and (c) in panel II of FIG. 22 ), and theCMTF-based (points I through IV, and point V, in FIG. 21 ). Such anillustration is provided in FIG. 25 . In the figure, the dashed lines(and the respective cross-hatched areas) correspond to the “ideal”signal of interest (without noise and adjacent channel interference),and the solid lines correspond to the signal+noise+interferencemixtures. The leftmost panels show the time domain traces, the rightmostpanels show the power spectral densities (PSDs), and the middle panelsshow the amplitude densities. The baseband power of the AWGN is onetenth of that of the signal of interest (10 dB AWGN SNR), and thebaseband power of the impulsive noise is approximately 8 times (9 dB)that of the AWGN. The value of the parameter β for Tukey’s range is β =3. (These noise and adjacent channel interference conditions, and thevalue β = 3, correspond to the respective channel capacities marked bythe asterisks in FIG. 24 ).

Measure of peakedness - In the panels showing the amplitude densities,the peakedness of the signal+noise mixtures is measured and indicated inunits of “decibels relative to Gaussian” (dBG). This measure is based onthe classical definition of kurtosis [50], and for a real-valued signalmay be expressed in terms of its kurtosis in relation to the kurtosis ofthe Gaussian (aka normal) distribution as follows [9, 10]:

$\begin{matrix}{K_{\text{dBG}}(x) = 101\text{g}\left\lbrack \frac{\left\langle \left( {x - \left\langle x \right\rangle} \right)^{4} \right\rangle}{3\left\langle \left( {x - \left\langle x \right\rangle} \right)^{2} \right\rangle^{2}} \right\rbrack,} & \text{­­­(35)}\end{matrix}$

where the angular brackets denote the time averaging. According to thisdefinition, a Gaussian distribution would have zero dBG peakedness,while sub-Gaussian and super-Gaussian distributions would have negativeand positive dBG peakedness, respectively. In terms of the amplitudedistribution of a signal, a higher peakedness compared to a Gaussiandistribution (super-Gaussian) normally translates into “heavier tails”than those of a Gaussian distribution. In the time domain, highpeakedness implies more frequent occurrence of outliers, that is, animpulsive signal.

Incoming signal - As one may see in the upper row of panels in FIG. 25 ,the incoming impulsive noise dominates over the AWGN. The peakedness ofthe signal+noise mixture is high (14.9 dBG), and its amplitudedistribution has a heavy “tail” at positive amplitudes.

Linear chain - The anti-aliasing filter in the linear chain (row (b))suppresses the high-frequency content of the noise, reducing thepeakedness to 2.3 dBG. The matching filter in the baseband (row (c))further limits the noise frequencies to within the baseband, reducingthe peakedness to 0 dBG. Thus the observed baseband noise may beconsidered to be effectively Gaussian, and we may use the Shannonformula [44] based on the achieved baseband SNR (0.9 dB) to calculatethe channel capacity. This is marked by the asterisk on the respectivesolid curve in FIG. 24 . (Note that the achieved 0.9 dB baseband SNR isslightly larger than the “ideal” 0.5 dB SNR that would have beenobserved without “clipping” the outliers of the output of theanti-aliasing filter by the A/D at ±V_(c).)

CMTF-based chain - As one may see in the panels of row V, the differencesignal largely reflects the temporal and the amplitude structures of thenoise and the adjacent channel signal. Thus its output may be used toobtain the range for identifying the noise outliers (i.e. the blankingvalue V_(c)/g). Note that a slight increase in the peakedness (from 14.9dBG to 15.4 dBG) is mainly due to decreasing the contribution of theGaussian signal of interest, as follows from the linearity property ofkurtosis.

As may be seen in the panels of row II, since the CMTFdisproportionately affects signals with different temporal and/oramplitude structures, it reduces the spectral density of the impulsiveinterference in the signal passband without significantly affecting thesignal of interest. The impulsive noise is notably decreased, while theamplitude distribution of the filtered signal+noise mixture becomeseffectively Gaussian.

The anti-aliasing (row III) and the baseband (row IV) filters furtherreduce the remaining noise to within the baseband, while the modifiedbaseband filter also compensates for the insertion of the CMTF in thesignal chain. This results in the 9.3 dB baseband SNR, leading to thechannel capacity marked by the asterisk on the respective dashed-linecurve in FIG. 24 .

4.4 Alternative Topology for Signal Processing Chain Shown in FIG. 21

FIG. 26 provides an illustration of an alternative topology for signalprocessing chain shown in FIG. 21 , where the blanking range isdetermined according to equation (67).

One skilled in the art will recognize that the topology shown in FIG. 21and the topology shown in FIG. 26 both comprise a CMTF filter(transforming an input signal x(t) into an output signal χ(t))characterized by a blanking range, and a robust means to establish saidblanking range in such a way that it excludes the outliers in thedifference signal x(t) - χ(t).

In FIG. 26 , two QTFs receive a signal proportional to the blanker inputand produce two QTF outputs, corresponding to the 1st and the 3rdquartiles of the QTF input. Then the blanking range of the blanker isestablished as a linear combination of these two outputs.

5 ΔΣ ADC With CMTF-Based Loop Filter

While §4 discloses mitigation of outlier noise in the process ofanalog-to-digital conversion by ADiCs/CMTFs deployed ahead of an ADC,CMTF-based outlier noise filtering of the analog input signal may alsobe incorporated into loop filters of ΔΣ analog-to-digital converters.

Let us consider the modifications to a 2nd-order ΔΣ ADC depicted in FIG.27 . (Note that the vertical scales of the shown fragments of the signaltraces vary for different fragments.) We may assume from here on thatthe 1st order lowpass filters with the time constant τ and the impulseresponse h_(τ)(t) shown in the figure have a bandwidth (as signified bythe 3 dB corner frequency) that is much larger than the bandwidth of thesignal of interest B_(x), yet much smaller than the sampling (clock)frequency F_(s). For example, the bandwidth of h_(τ)(t) may beapproximately equal to the geometric mean of B_(x) and F_(s), resultingin the following value for τ:

$\begin{matrix}{\tau \approx \frac{1}{2\pi\sqrt{B_{x}F_{\text{s}}}}.} & \text{­­­(36)}\end{matrix}$

As one may see in FIG. 27 , the first integrator (with the time constantγτ) is preceded by a (symmetrical) blanker, where the (symmetrical)blanking function B_(α)(x) may be defined as

$\begin{matrix}{B_{\alpha}(x) = \left\{ \begin{array}{l}{x\quad\text{for}\quad|x| \leq \alpha} \\{0\quad\text{otherwise}}\end{array} \right),} & \text{­­­(37)}\end{matrix}$

and where α is the blanking value.

As shown in the figure, the input x(t) and the output y(t) may berelated by

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\overline{h_{\tau}\ast y} = \frac{1}{\gamma\tau}\overline{B_{\alpha}\left( {h_{\tau}\ast\left( {x-y} \right)} \right)},} & \text{­­­(38)}\end{matrix}$

where the overlines denote averaging over a time interval between anypair of threshold (including zero) crossings of D (such as, e.g., theinterval ΔT shown in FIG. 27 ), and the filter represented by equation(38) may be referred to as a Clipped Mean Tracking Filter (CMTF). Notethat without the time averaging equation (38) corresponds to the ABAINFdescribed by equation (15) with µ = 0, where x and χ replaced by h_(τ)∗xand h_(τ)∗y, respectively.

The utility of the 1st order lowpass filters h_(τ)(t) would be, first,to modify the amplitude density of the difference signal x - y so thatfor a slowly varying signal of interest x(t) the mean and the medianvalues of h_(τ) ∗ (x - y) in the time interval ΔT would becomeeffectively equivalent, as illustrated in FIG. 28 . However, the medianvalue of h_(τ) ∗ (x - y) would be more robust when the narrow-bandsignal of interest is affected by short-duration outliers such asbroadband impulsive noise, since such outliers would not be excessivelybroadened by the wide-band filter h_(τ)(t). In addition, while beingwide-band, this filter would prevent the amplitude of the backgroundnoise observed at the input of the blanker from being excessively large.

With τ given by equation (36), the parameter γ may be chosen as

$\begin{matrix}{\gamma \approx \frac{1}{4\pi B_{x}\tau} \approx \frac{1}{2}\sqrt{\frac{F_{\text{s}}}{B_{x}}},} & \text{­­­(39)}\end{matrix}$

and the relation between the input and the output of the ΔΣ ADCs with aCMTF-based loop filter may be expressed as

$\begin{matrix}{x\left( {t - \text{Δ}t} \right) \approx \left( {\left( {w + \gamma\tau\overset{˙}{w}} \right) \ast y} \right)(t).} & \text{­­­(40)}\end{matrix}$

Note that for large blanking values such that α ≥ |h_(τ)∗(x-y)| for allt, according to equation (38) the average rate of change of h_(τ)∗ywould be proportional to the average of the difference signalh_(τ)∗(x-y). When the magnitude of the difference signal h_(τ) ∗ (x-y)exceeds the blanking value α, however, the average rate of change ofh_(τ)∗y would be zero and would no longer depend on the magnitude ofh_(τ)∗x, providing an output that would be insensitive to outliers witha characteristic amplitude determined by the blanking value α.

Since linear filters are generally better than median for removingbroadband Gaussian (e.g. thermal) noise, the blanking value in theCMTF-based topology should be chosen to ensure that the CMTF-based ΔΣADC performs effectively linearly when outliers are not present, andthat it exhibits nonlinear behavior only intermittently, in response tooutlier noise. An example of a robust approach to establishing such ablanking value is outlined in §5.2.

One skilled in the art will recognize that the ΔΣ modulator depicted inFIG. 27 comprises a quantizer (flip-flop), a blanker, two integrators,and two wide-bandwidth 1st order lowpass filters, and the (nonlinear)loop filter of this modulator is configured in such a way that the valueof the quantized representation (signal y(t)) of the input signal x(t),averaged over a time interval ΔT comparable with an inverse of thenominal bandwidth of the signal of interest, is effectively proportionalto a nonlinear measure of central tendency of said input signal x(t) insaid time interval ΔT.

5.1 Simplified Performance Example

Let us first use a simplified synthetic signal to illustrate theessential features, and the advantages provided by the ΔΣ ADC with theCMTF-based loop filter configuration when the impulsive noise affectingthe signal of interest dominates over a low-level background Gaussiannoise.

In this example, the signal of interest consists of two fragments of twosinusoidal tones with 0.9 V_(c) amplitudes, and with frequencies B_(x)and B_(x)/8, respectively, separated by zero-value segments. While puresine waves are chosen for an ease of visual assessment of the effects ofthe noise, one may envision that the low-frequency tone corresponds to avowel in a speech signal, and that the high-frequency tone correspondsto a fricative consonant.

For all ΔΣ ADCs in this illustration, the flip-flop clock frequency isF_(s) = NB_(x), where N = 1024. For the 2nd-order loop filter in thisillustration τ = (4πB_(x))⁻¹. The time constant τ of the 1st orderlowpass filters in the CMTF-based loop filter is

$\tau = \left( {2\pi B_{x}\sqrt{N}} \right)^{- 1} = \left( {64\pi B_{x}} \right)^{- 1},$

and γ = 16 (resulting in γτ = (4πB_(x))⁻¹). The parameter α is chosen asα = V_(c). The output y[k] of the ΔΣ ADC with the 1st-order linear loopfilter (panel I of FIG. 1 ) is filtered with a digital lowpass filterwith the impulse response w[k]. The outputs of the ΔΣ ADCs with the2nd-order linear (panel II of FIG. 1 ) and the CMTF-based (FIG. 5 ) loopfilters are filtered with a digital lowpass filter with the impulseresponse w[k] + (4πB_(x))⁻¹ẇ[k]. The impulse and frequency responses ofw[k] and w[k] + (4πB_(x))⁻¹ẇ[k] are shown in FIG. 29 .

As shown in panel I of FIG. 30 , the signal is affected by a mixture ofadditive white Gaussian noise (AWGN) and white impulse (outlier) noisecomponents, both band-limited to approximately

$B_{x}\sqrt{N}$

bandwidth. As shown in panel II, in the absence of the outlier noise,the performance of all ΔΣ ADC in this example is effectively equivalent,and the amount of the AWGN is such that the resulting signal-to-noiseratio for the filtered output is approximately 20 dB in the absence ofthe outlier noise. The amount of the outlier noise is such that theresulting signal-to-noise ratio for the filtered output of the ΔΣ ADCwith a 1st-order linear loop filter is approximately 6 dB in the absenceof the AWGN.

As one may see in panels III and IV of FIG. 30 , the linear loop filtersare ineffective in suppressing the impulsive noise. Further, theperformance of the ΔΣ ADC with the 2nd-order linear loop filter (seepanel IV of FIG. 30 ) is more severely degraded by high-power noise,especially by high-amplitude outlier noise such that the condition|x(t - Δt) + (w ∗ v)(t)| < V_(c) is not satisfied for all t. On theother hand, as may be seen in panel V of FIG. 30 , the ΔΣ ADC with theCMTF-based loop filter improves the signal-to-noise ratio by about 13 dBin comparison with the ΔΣ ADC with the 1st-order linear loop filter,thus removing about 95% of the impulsive noise.

More importantly, as may be seen in panel III of FIG. 31 , increasingthe impulsive noise power by an order of magnitude hardly affects theoutput of the ΔΣ ADC with the CMTF-based loop filter (and thus about99.5% of the impulsive noise is removed), while further exceedinglydegrading the output of the ΔΣ ADC with the 1st-order linear loop filter(panel II).

5.2 ΔΣ ADC With Adaptive CMTF

A CMTF with an adaptive (possibly asymmetric) blanking range [α₋, α₊]may be designed as follows. To ensure that the values of the differencesignal h_(τ)∗(x-y) that lie outside of [a_, a₊] are outliers, one mayidentify [a_, a₊] with Tukey’s range [48], a linear combination of the1st (Q₁) and the 3rd (Q₃) quartiles of the difference signal (see [33,34] for additional discussion of quantiles of continuous signals):

$\begin{matrix}{\left\lbrack {\alpha_{-},\alpha_{+}} \right\rbrack = \left\lbrack {Q_{1} - \beta\left( {Q_{3} - Q_{1}} \right),Q_{3} + \beta\left( {Q_{3} - Q_{1}} \right)} \right\rbrack,} & \text{­­­(41)}\end{matrix}$

where β is a coefficient of order unity (e.g. β = 1.5). From equation(41), for a symmetrical distribution the range that excludes outliersmay also be obtained as [α₋, α₊] = [-α, α], where α is given by

$\begin{matrix}{\alpha = \left( {1 + 2\beta} \right)Q_{2}^{\star},} & \text{­­­(42)}\end{matrix}$

and where

Q₂^(⋆)

is the 2nd quartile (median) of the absolute value (or modulus) of thedifference signal |h_(τ)∗(x-y)|.

Alternatively, since

2Q₂^(⋆) = Q₃ − Q₁

for a symmetrical distribution, the resolution parameter α may beobtained as

$\begin{matrix}{\alpha = \left( {\frac{1}{2} + \beta} \right)\left( {Q_{3} - Q_{1}} \right),} & \text{­­­(43)}\end{matrix}$

where Q₃ - Q₁ is the interquartile range (IQR) of the difference signal.

FIG. 32 provides an outline of a ΔΣ ADC with an adaptive CMTF-based loopfilter. In this example, the 1st order lowpass filters are followed bythe gain stages with the gain G, while the blanking value is set toV_(c). Note that

$\begin{matrix}{B_{V_{c}}\left( {Gx} \right) = GB_{\frac{V_{c}}{G}}(x),} & \text{­­­(44)}\end{matrix}$

and thus the “apparent” (or “equivalent”) blanking value would be nolonger hardware limited. As shown in FIG. 32 , the input x(t) and theoutput y(t) may be related by

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\overline{h_{\tau}\ast y} = \frac{1}{\gamma\tau}\overline{B_{\frac{V_{c}}{G}}\left( {h_{\tau}\ast\left( {x-y} \right)} \right)}.} & \text{­­­(45)}\end{matrix}$

If an automatic gain control circuit maintains a constant output-V_(c)/(1 + 2β) of the MTF circuit in FIG. 32 , then the apparentblanking value α = V_(c)/G in equation (45) may be given by equation(42).

5.2.1 Performance Example

Simulation parameters - To emulate the analog signals in the examplesbelow, the digitization rate is two orders of magnitude higher than thesampling rate F_(s). The signal of interest is a Gaussian basebandsignal in the nominal frequency rage [0, B_(x)]. It is generated as abroadband white Gaussian noise filtered with a root-raised-cosine filterwith the roll-off factor ¼ and the bandwidth 5B_(x)/4. The noiseaffecting the signal of interest is a sum of an AWGN backgroundcomponent and white impulsive noise i(t). The impulsive noise is modeledas symmetrical (bipolar) Poisson shot noise:

$\begin{matrix}{i(t) = v(t){\sum\limits_{k = 1}^{\infty}{\delta\left( {t - t_{k}} \right)}},} & \text{­­­(46)}\end{matrix}$

where v(t) is AWGN noise, t_(k) is the k-th arrival time of a Poissonprocess with the rate parameter λ, and δ(x) is the Dirac δ-function[31]. In the examples below, λ = B_(x). The gain G is chosen to maintainthe output of the MTF in FIG. 32 at - V_(c)/(1+2β), and the digitallowpass filter w[k] is the root-raised-cosine filter with the roll-offfactor ¼ and the bandwidth 5B_(x)/4. The remaining hardware parametersare the same as those in §5.1. Further, the magnitude of the input x(t)is chosen to ensure that the average absolute value of the output signalis approximately V_(c)/5.

Comparative channel capacities - For the simulation parameters describedabove, FIG. 33 compares the simulated channel capacities (calculatedfrom the baseband SNRs using the Shannon formula [44]) for varioussignal+noise compositions, for the linear signal processing chain (solidlines) and the CMTF-based chain of FIG. 32 with β = 1.5 (dotted lines).

As one may see in FIG. 33 (and compare with the simplified diagram ofFIG. 2 ), linear and the CMTF-based chains provide effectivelyequivalent performance when the AWGN dominates over the impulsive noise.However, the CMTF-based chains are insensitive to further increase inthe impulsive noise when the latter becomes comparable or dominates overthe thermal (Gaussian) noise, thus providing resistance to impulsiveinterference.

Disproportionate effect on baseband PSDs - For a mixture of whiteGaussian and white impulsive noise, FIG. 34 illustrates reduction of thespectral density of impulsive noise in the signal baseband withoutaffecting that of the signal of interest. In the figure, the solid linescorrespond to the “ideal” signal of interest (without noise), and thedotted lines correspond to the signal+noise mixtures. The baseband powerof the AWGN is one tenth of that of the signal of interest (10 dB AWGNSNR), and the baseband power of the impulsive noise is approximately 8times (9 dB) that of the AWGN. The value of the parameter β for Tukey’srange is β = 1.5. As may be seen in the figure, for the CMTF-based chainthe baseband SNR increases from 0.5 dB to 9.7 dB.

For both the linear and the CMTF-based chains the observed basebandnoise may be considered to be effectively Gaussian, and we may use theShannon formula [44] based on the achieved baseband SNRs to calculatethe channel capacities. Those are marked by the asterisks on therespective solid and dotted curves in FIG. 33 .

FIG. 35 provides a similar illustration with additional interference inan adjacent channel. Such interference increases the apparent blankingvalue needed to maintain effectively linear CMTF behavior in the absenceof the outliers, slightly reducing the effectiveness of the impulsivenoise suppression. As a result, the baseband SNR increases from 0.5 dBto only 8.5 dB.

6 ΔΣ ADCs With Linear Loop Filters and Digital ADiC/CMTF Filtering

While §5 describes CMTF-based outlier noise filtering of the analoginput signal incorporated into loop filters of ΔΣ analog-to-digitalconverters, the high raw sampling rate (e.g. the flip-flop clockfrequency) of a ΔΣ ADC (e.g. two to three orders of magnitude largerthan the bandwidth of the signal of interest) may be used for effectiveABAINF/CMTF/ADiC-based outlier filtering in the digital domain,following a ΔΣ modulator with a linear loop filter.

FIG. 36 shows illustrative signal chains for a ΔΣ ADC with linear loopand decimation filters (panel (a)), and for a ΔΣ ADC with linear loopfilter and ADiC-based digital filtering (panel (b)). As may be seen inpanel (a) of FIG. 36 , the quantizer output of a ΔΣ ADC with linear loopfilter would be filtered with a linear decimation filter that wouldtypically combine lowpass filtering with downsampling. To enable anADiC-based outlier filtering (panel (b)), a wideband (e.g. withbandwidth approximately equal to the geometric mean of the nominalsignal bandwidth B_(x) and the sampling frequency F_(s)) digital filteris first applied to the output of the quantizer. The output of thisfilter is then filtered by a digital ADiC (with appropriately chosentime parameter and the blanking range), followed by a linearlowpass/decimation filter.

FIG. 37 shows illustrative time-domain traces at points I through VI ofFIG. 36 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise). Inthis example, a 1st order ΔΣ modulator is used, and the quantizerproduces a 1-bit output shown in panel II. The digital wideband filteris a 2nd order IIR Bessel filter with the corner frequency approximatelyequal to the geometric mean of the nominal signal bandwidth B_(x) andthe sampling frequency F_(s). The time parameter of the ADiC isapproximately τ ≈ (4πB_(x))⁻¹, and the same lowpass/decimation filter isused as for the linear chain of FIG. 36 (a).

FIG. 38 shows illustrative signal chains for a ΔΣ ADC with linear loopand decimation filters (panel (a)), and for a ΔΣ ADC with linear loopfilter and CMTF-based digital filtering (panel (b)). To enable aCMTF-based outlier filtering (panel (b)), a wideband digital filter isfirst applied to the output of the quantizer. The output of this filteris then filtered by a digital CMTF (with the time constant τ andappropriately chosen blanking range), followed by a linear lowpassfiltering (with the modified impulse response w[k] + τẇ[k]) combinedwith decimation.

FIG. 39 shows illustrative time-domain traces at points I through VI ofFIG. 38 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise). Inthis example, a 1st order ΔΣ modulator is used, and the quantizerproduces a 1-bit output shown in panel II. The digital wideband filteris a 2nd order IIR Bessel filter with the corner frequency approximatelyequal to the geometric mean of the nominal signal bandwidth B_(x) andthe sampling frequency F_(s). The time parameter of the CMTF is τ =(4πB_(x))⁻¹, and the impulse response of the lowpass filter in thedecimation stage is modified as w[k] + (4πB_(x))⁻¹ ẇ[k].

To prevent excessive distortions of the quantizer output byhigh-amplitude transients (especially for high-order ΔΣ modulators), andthus to increase the dynamic range of the ADC and/or the effectivenessof outlier filtering, an analog clipper (with appropriately chosenclipping values) should precede the ΔΣ modulator, as schematically shownin FIGS. 40 and 42 .

FIG. 40 shows illustrative signal chains for a ΔΣ ADC with linear loopand decimation filters (panel (a)), and for a ΔΣ ADC with linear loopfilter and ADiC-based digital filtering (panel (b)), with additionalclipping of the analog input signal.

FIG. 41 shows illustrative time-domain traces at points I through VI ofFIG. 40 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise). Inthis example, a 1st order ΔΣ modulator is used, and the quantizerproduces a 1-bit output. The digital wideband filter is a 2nd order IIRBessel filter with the corner frequency approximately equal to thegeometric mean of the nominal signal bandwidth B_(x) and the samplingfrequency F_(s). The time parameter of the ADiC is approximately τ ≈(4πB_(x))⁻¹, and the same lowpass/decimation filter is used as for thelinear chain of FIG. 40 (a).

FIG. 42 shows illustrative signal chains for a ΔΣ ADC with linear loopand decimation filters (panel (a)), and for a ΔΣ ADC with linear loopfilter and CMTF-based digital filtering (panel (b)), with additionalclipping of the analog input signal.

FIG. 43 shows illustrative time-domain traces at points I through VI ofFIG. 42 , and the output of the ΔΣ ADC with linear loop and decimationfilters for the signal affected by AWGN only (w/o impulsive noise). Inthis example, a 1st order ΔΣ modulator is used, and the quantizerproduces a 1-bit output. The digital wideband filter is a 2nd order IIRBessel filter with the corner frequency approximately equal to thegeometric mean of the nominal signal bandwidth B_(x) and the samplingfrequency F_(s). The time parameter of the CMTF is τ = (4πB_(x))⁻¹, andthe impulse response of the lowpass filter in the decimation stage ismodified as w[k] + (4πB_(x))⁻¹ ẇ[k].

7 ADiC Variants

Let us revisit the ADiC block diagram shown in FIG. 12 , where thedepreciator is a blanker. In FIG. 12 , x(t) is the input, y(t) is theoutput, and the “intermediate” signal χ(t) may be called theDifferential Clipping Level (DCL). Without the blanker (or outliers),the DCL would be the output of a 1st order linear lowpass filter withthe corner frequency 1/(2πτ). The blanker input is the “differencesignal” x(t) - χ(t). The blanker output would be equal to its input ifthe input falls within the blanking range, and would be zero otherwise.

If there is established a robust range [a_, α₊] around the differencesignal, then whatever protrudes from this range may be identified as anoutlier. As has been previously shown in this disclosure, such a robustrange may be established in real time, for example, using quantiletracking filters.

While in the majority of the examples in this disclosure a robust rangeis established using quantile tracking filters, one skilled in the artwill recognize that such a range may also be established based on avariety of other robust measures of dispersion of the difference signal,such as, for example, mean or median absolute deviation.

Here (and throughout the disclosure) “robust” should be read as“insensitive to outliers” when referred to filtering, establishing arange, estimating a measure of central tendency, etc.

“Robust” may also be read as “less-than-proportional” when referred tothe change in an output of a filter, an estimator of a range and/or of ameasure of central tendency, etc., in response to a change in theamplitude and/or the power of outliers.

While a linear filter (e.g. lowpass, bandpass, or bandstop) may not be arobust filter in general, it may perform as a robust filter when appliedto a mixture of a signal and outliers when the signal and the outliershave sufficiently different bandwidths. For example, consider a mixtureof a band-limited signal of interest and a wideband impulsive noise, anda linear filter that is transparent to the signal of interest whilebeing opaque to the frequencies outside of the signal’s band. When sucha filter is applied to such a mixture, the amplitude and/or power of thesignal of interest would not be affected, while the amplitude and/orpower of the outliers (i.e. the impulsive noise) would be reduced. Thusthis linear filter, while affecting the PSD of both the signal and theimpulsive noise proportionally in the filter’s passband, woulddisproportionately affect their PSDs outside of the filter’s passband,and would disproportionately affect their amplitudes.

When the blanker’s output is zero (that is, according to the abovedescription, an outlier is encountered), the DCL χ(t) in the ADiC shownin FIG. 12 would be maintained at its previous level. As the result, inthe ADiC’s output the outliers would be replaced by the DCL χ(t),otherwise the signal would not be affected. Thus the ADiC’s output wouldremain bounded to within the blanking (or transparency) range around theDCL.

As discussed in §2.4, a DCL may also be formed by the output of a robustMeasure of Central Tendency (MCT) filter such as, e.g., a CMTF, and theADiC output may be formed as a weighted average of the input signal andthe DCL (see equation (20)).

As discussed above (and especially when the signal and the outliers havesufficiently different bandwidths), a DCL may also be formed by theoutput of a linear filter that disproportionately reduces the amplitudesof the outliers in comparison with that of the signal of interest. An“ideal” linear filter to establish such a DCL would be a filter havingan effectively unity frequency response and an effectively zero groupdelay over the bandwidth of the signal of interest.

When applied to the input signal x(t) comprising a signal of interest, alinear filter having an effectively unity frequency response and aneffectively constant group delay Δt > 0 over the bandwidth of the signalof interest would establish a DCL for a delayed signal x(t-Δt).

Further, a DCL may be formed by a large variety of linear and/ornonlinear filters, such that a filter produces an output that representsa measure of location of the input signal in a moving time window (aWindowed Measure of Location, or WML), and/or by a combination of suchfilters.

Thus, as illustrated in FIG. 44 , an ADiC structure may also be giventhe following alternative description.

First, a Differential Clipping Level (DCL) χ(t) is formed. In FIG. 44 ,the DCL is formed by a DCL circuit that converts the input signal x(t)into the output χ(t). A preferred way to establish such a DCL would beto obtain it as an output of a robust (i.e. insensitive to outliers)filter estimating a local (windowed) measure of location (a WindowedMeasure of Location, or WML) of the input signal x(t). Such a filter maybe, for example, a median filter, a CMTF, an NDL, an MTF, a TrimeanTracking Filter (TTF), or a combination of such filters. However, a DCLmay also be obtained by a linear (e.g. lowpass, bandpass, or bandstop)filter applied to the input signal x(t), or by a combination of linearand robust nonlinear filters mentioned above.

Then, a difference signal x(t) - χ(t) is obtained as the differencebetween the input signal x(t) and the DCL χ(t).

Next, a robust range [α₋(t), α₊(t)] of the difference signal isdetermined, by a Robust Range Circuit (RRC), as a range between theupper (α₊(t)) and the lower (α₋(t)) robust “fences” for the differencesignal. For example, such fences may be constructed as linearcombinations of the outputs of quantile tracking filters, includinglinear combinations of the outputs of quantile tracking filters withdifferent slew rate parameters. Several examples of (analog and/ordigital) RRCs are provided in this disclosure, including those shown inFIGS. 21, 26, 32, 51, and 52 .

The difference signal and the fences are used as input signals of adepreciator (or a differential depreciator, as described below)characterized by an influence function (or a differential influencefunction having a difference response, as described below) and producinga depreciator output that is effectively equal to the difference signalwhen the difference signal is within the robust range [α₋(t), α₊(t)](the “blanking range”, or “transparency range”), smaller than thedifference signal when the difference signal is larger than α₊(t), andlarger than the difference signal when the difference signal is smallerthan α₋(t).

In the examples of the depreciators discussed above, the influencefunction

I_(α⁻)^(α₊)(x)

of a depreciator is characterized by the transparency function

T_(α⁻)^(α₊)(x)

such that

I_(α⁻)^(α₊)(x) = xT_(α⁻)^(α₊)(x)

(see, e.g., equations (12), (13), and (14)), and thus those examplesimply that α₋(t) < 0 < α₊(t). Various examples of such transparencyfunctions are given throughout the disclosure, including those shown inFIGS. 3, 4, 20, 57, 59, 80, and 82 .

In order to efficiently depreciate outliers when sign(α₋(t)) =sign(α₊(t)), it may be preferred to use a differential depreciator. Thedifferential influence function

$\overline{I_{\alpha_{-}}^{\alpha_{+}}}(x)$

of such a differential depreciator may be related to the influencefunction

I_(α⁻)^(α₊)(x)

of the depreciators discussed previously as follows:

$\begin{matrix}{\overline{I_{\alpha_{-}}^{\alpha_{+}}}(x) = M + I_{\alpha_{-} - M}^{\alpha_{+} - M}\left( {x - M} \right),} & \text{­­­(47)}\end{matrix}$

where

ℳ

is an average value of the lower and the upper fences,

α_(t) < ℳ < α₊(t)

(e.g.ℳ = (α_(t) + α₊(t))/2).

Note that it follows from equation (47) and the above discussion ofinfluence functions that

$x < \overline{I_{\alpha_{-}}^{\alpha_{+}}}(x) \leq M\text{for}x < \alpha_{-}\text{and}M \leq \overline{I_{\alpha_{-}}^{\alpha_{+}}}(x) < x\text{for}\alpha_{+} < x.$

It may be convenient to characterize a differential depreciator with thedifferential influence function

$\overline{I_{\alpha_{-}}^{\alpha_{+}}}(x)$

by its difference response

$x - \overline{I_{\alpha_{-}}^{\alpha_{+}}}(x)$

(i.e. by the difference between the input and the output of adepreciator), as illustrated in FIG. 45 . It may be seen in FIG. 45 thatthe difference response of a differential depreciator is a monotonicallyincreasing function of its input, and the difference response

$x - \overline{I_{\alpha_{-}}^{\alpha_{+}}}(x)$

is effectively zero when the depreciator input is within thetransparency range (i.e. for α₋ < x < α₊).

A function f(x) would be monotonically increasing (also increasing ornon-decreasing) if for any Δx ≥ 0 f (x+Δx) ≥ f(x).

Finally, as shown in FIG. 44 , the ADiC output y(t) is formed as a sumof the DCL χ(t) and the depreciator output.

Specifically, for the blanking influence function

B_( α⁻)^(α₊)(x)

(e.g. given by equation (18)), the ADiC output y(t) would beproportional to the ADiC input x(t) when the difference signal is withinthe range [a_, α₊], and it would be proportional to the DCL χ(t)otherwise:

$\begin{matrix}{y = G\left\lbrack {\chi + B_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}\left( {x - \chi} \right)} \right\rbrack = G\left\{ \begin{matrix}x & {\text{­­­(48)}\alpha_{-} \leq x - \chi \leq \alpha_{+}} \\\chi & \text{otherwise}\end{matrix} \right)\mspace{6mu},} & \end{matrix}$

where G is a positive or a negative gain value.

FIG. 46 shows a block diagram of such an ADiC with a blankingdepreciator, where the multiplication of the ADiC input x(t) and the DCLχ(t) by the depreciator weights is performed by a single poledouble-throw switch (SPDT), forming the ADiC output y(t) according toequation (48) with G = 1.

As shown in FIG. 46 , a Window Detector Circuit (WDC) (or WindowComparator Circuit (WCC), or Dual Edge Limit Detector Circuit (DELDC))may be used to determine whether the difference signal is within therange [a_, α₊], where said range is produced by an RRC.

In FIG. 46 , the WDC outputs a two-level Switch Control Signal (SCS) sothat the 1st level corresponds to the WDC input (the difference signalx(t) - χ(t)) being within the range [α₋, α₊], α₋ < x(t) - χ(t) < α₊, andotherwise the WDC outputs the 2nd level. The 1st level WDC output putsthe switch in position “1”, and the second level puts the switch inposition “2”. As the result, in the ADiC output y(t) the outliers wouldbe replaced by the DCL χ(t), otherwise the signal would not be affected.

FIG. 47 shows illustrative signal traces for the ADiC shown in FIG. 46with the DCL established by a linear lowpass filter, and the lower(α₊(t)) robust fences for the difference signal are constructed as alinear combination of the outputs of two QTFs. In this example, the WMLof the input signal x(t) is obtained as a weighted average in the movingwindow w(t).

If the outliers are depreciated by a differential blanker with theinfluence function

$\overline{B_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}}(x)$

given by

$\begin{matrix}{\overline{B_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}}(x) = G\left\{ \begin{matrix}x & {\text{­­­(49)}\alpha_{-} \leq x \leq \alpha_{+}} \\\frac{\alpha_{+} + \alpha_{-}}{2} & \text{otherwise}\end{matrix} \right)\mspace{6mu},} & \end{matrix}$

then the ADiC output y(t) would be given by

$\begin{matrix}{y(t) = G\left\{ \begin{matrix}{x(t)} & {\text{­­­(50)}\alpha_{-}(t) \leq x(t) - \chi(t) \leq \alpha_{+}(t)} \\{\chi(t) + \frac{\alpha_{+}(t) + \alpha_{-}(t)}{2}} & \text{otherwise}\end{matrix} \right),} & \end{matrix}$

where G is a positive or a negative gain value.

7.1 Robust Filters

While a linear filter (e.g. lowpass, bandpass, or bandstop) may not be arobust filter in general, it may perform as a robust filter when appliedto a mixture of a signal and outliers when the signal and the outliershave sufficiently different bandwidths. In such a case, a linear filter,while affecting the PSD of both the signal and the impulsive noiseproportionally in the filter’s passband, would disproportionately affecttheir PSDs outside of the filter’s passband, and woulddisproportionately affect their amplitudes.

Examples of nonlinear filters estimating a robust local measure oflocation of the input signal x(t) include, but are not limited to, thefollowing nonlinear filters: a median filter; a slew rate limitingfilter; a Nonlinear Differential Limiter (NDL) [9, 10, 24, 32]; aClipped Mean Tracking Filter (CMTF); a Median Tracking Filter (MTF); aTrimean Tracking Filter (TTF) described below (see §7.1.1).

7.1.1 Trimean Tracking Filter (TTF)

Simple yet efficient real-time robust filters may be constructed asweighted averages of outputs of quantile tracking filters described in§3.

In particular, a Trimean Tracking Filter (TTF) may be constructed as aweighted average of the outputs of the MTF (§3.1) and the QTFs (§3.2):

$\begin{matrix}{Q_{12\text{w3}}(t) = \frac{Q_{1}(t) + wQ_{2}(t) + Q_{3}(t)}{2 + w},} & \text{­­­(51)}\end{matrix}$

where w ≥ 0.

Note that in practical electronic-circuit (analog) TTF implementationscontinuous high-resolution comparators (see §11.3) may be used forimplementing the MTF and the QTFs. Alternatively, comparators withhysteresis (Schmitt triggers) may be used to reduce the comparatorswitching rates when the values of the inputs of the MTF and the QTFsare close to their respective outputs.

An example of a numerical algorithm implementing a finite-differenceversion of a TTF may be given by the following MATLAB function:

function y = TTF(x,dt,mu,w) y = zeros(size(x(:)));gamma = mu∗dt; Q3 = x(1); Q2 = x(1); Q1 = x(1); y(1) = x(1);for i = 2:length(x);     dX = x(i) - Q3;    if dX > -0.5∗gamma & dX < 1.5∗gamma. Q3 = x(i);    else Q3 = Q3 + gamma∗(sign(dX)+0.5);     end     dX = x(i) - Q2;    if abs(dX) < gamma. Q2 = x(i);     else. Q2 = Q2 + gamma∗sign(dX);    end     dX = x(i) - Q1;    if dX > -1.5∗gamma & dX < 0.5∗gamma. Q1 = x(i);    else. Q1 = Q1 + gamma∗(sign(dX)-0.5);     end    y(i) = (Q1+w∗Q2+Q3)/(2+w); end return

An example of a numerical algorithm implementing a numerical version ofan ADiC with the DCL formed by a TTF may be given by the followingMATLAB function:

function [xADiC,xDCL,alpha_p,alpha_m] = ADiC_TTF(x,dt,mu_TTF,w,mu_range,beta)%------------------------------------------------------------------------------xADiC = zeros(size(x)); xDCL = zeros(size(x));alpha_p = zeros(size(x)); alpha_m = zeros(size(x));gamma_TTF = mu_TTF∗dt; gamma_range = mu_range∗dt;xADiC(1) = x(1); xDCL(1) = x(1);Q3 = x(1); Q2 = x(1); Q1 = x(1); dQ3 = 0; dQ1 = 0;%------------------------------------------------------------------------------for i = 2:length(x); % TRIMEAN TRAKING FILTER (TTF)     dX = x(i) - Q3;    if dX > -0.5∗gamma_TTF & dX < 1.5∗gamma_TTF Q3 = x(i);    else Q3 = Q3 + gamma_TTF∗(sign(dX)+0.5); end     dX = x(i) - Q2;    if abs(dX) < gamma_TTF Q2 = x(i);    else Q2 = Q2 + gamma_TTF∗sign(dX); end     dX = x(i) - Q1;    if dX > -1.5∗gamma_TTF & dX < 0.5∗gamma_TTF Q1 = x(i);    else Q1 = Q1 + gamma_TTF∗(sign(dX)-0.5); end % TRIMEAN DCL    xDCL(i) = (Q1+w∗Q2+Q3)/(2+w); % “Difference Signal”    dX = x(i) - xDCL(i);% QUARTILE TRACKING FILTERS (QTFs) for difference signal    dX3 = dX - dQ3;    if dX3 > -0.5∗gamma_range & dX3 < 1.5∗gamma_range dQ3 = dX;    else dQ3 = dQ3 + gamma_range∗(sign(dX3)+0.5); end    dX1 = dX - dQ1; if dX1 > -1.5∗gamma_range & dX1 < 0.5∗gamma_range dQ1 = dX;    else dQ1 = dQ1 + gamma_range∗(sign(dX1)-0.5); end % TUKEY’S RANGE    alpha_p(i) = dQ3 + beta∗(dQ3-dQ1);    alpha_m(i) = dQ1 - beta∗(dQ3-dQ1); % ADiC output    if dX>alpha_p(i) | dX<alpha_m(i) xADiC(i) = xDCL(i);    else xADiC(i) = x(i); end end return

FIG. 48 and FIG. 49 show illustrative signal traces for the ADiC shownin FIG. 46 with the DCL established by a TTF.

The top panel in FIG. 48 shows the input ADiC signal without noise (anup-chirp signal), and the panel second from the top shows the input ADiCsignal with noise (the signal x(t) at point I in FIG. 46 ). The middlepanel in FIG. 48 shows the DCL signal χ(t) established by a robustfilter (a TTF) applied to x(t) (the signal χ(t) at point II in FIG. 46), and the panel second from the bottom shows the difference signalx(t) - χ(t) (the signal at point III in FIG. 46 ). The bottom panel inFIG. 48 shows the upper (α₊(t)) and lower (α₋(t)) fences of thedifference signal produced by the Robust Range Circuit (RRC).

The top panel in FIG. 49 shows the bandpass-filtered up-chirp signalwithout noise, and the panel second from the top shows thebandpass-filtered signal with noise (the signal at point IV in FIG. 46). The middle panel in FIG. 49 shows the bandpass-filtered ADiC outputy(t) (the signal at point V in FIG. 46 ), while the lower two panelsshow the bandpass-filtered noise with and without ADiC.

8 Simplified ADiC Structure

Note that the robust fences α₊(t) and α₋(t) may be constructed for theinput signal itself (as opposed to the difference signal) in such a waythat the DCL value may be formed as an average M(t) of the upper andlower fences, e.g., as the arithmetic mean of the fences: χ(t) = M(t) =[α₊(t)+α₋(t)]/2. Then, if the depreciator in FIG. 44 is characterized bythe influence function

I_(α⁻)^(α₊)(x), whereα^(′)₊ = α₊ − Mandα^(′)⁻ = α⁻ − M,

the ADiC output y(t) would be described by

$\begin{matrix}{y = M + I_{\alpha_{-} - M}^{\alpha_{+} + M}\left( {x - M} \right) = \overline{I_{\alpha_{-}}^{\alpha_{+}}}(x).} & \text{­­­(52)}\end{matrix}$

FIG. 50 provides a block diagram of such simplified ADiC structurewherein the robust fences α₊(t) and α₋(t) are constructed for the inputsignal itself (as opposed to the difference signal) and the outliers aredepreciated by a differential depreciator

$\overline{I_{\alpha_{-}}^{\alpha_{+}}}(x).$

The robust fences α₊(t) and α₋(t) may be constructed in a variety ofways, e.g. as linear combinations of the outputs of QTFs applied to theinput signal.

FIG. 51 provides an example of such simplified ADiC structure whereinthe robust fences α₊(t) and α₋(t) are constructed for the input signalas linear combinations of the outputs of QTFs applied to the inputsignal, and wherein the outliers are depreciated by a differentialblanker that may be described by the differential blanking function

$\overline{B_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}}(x)$

given by equation (49).

FIG. 52 provides a simplified diagram of illustrative electronic circuitfor the ADiC structure shown in FIG. 51 . In this example, ADiC circuitacts as an Outlier-Removing Buffer (ORB), removing the outliers withoutotherwise affecting the signal. As an example, FIG. 53 providesillustrative signal traces from an LTspice simulation of such simple ORBcircuit. The upper panel in the figure shows the ORB’s input, the middlepanel shows its “fences” α₊(t) and α₋(t), and the lower panel shows theORB output.

An example of a numerical algorithm implementing a numerical version ofan ADiC shown in FIG. 51 may be given by the following MATLAB function:

function [xADiC,alpha_p,alpha_m] = ADiC_QTFs(x,dt,mu,beta)%------------------------------------------------------------------------------xADiC = zeros(size(x));alpha_p = zeros(size(x)); alpha_m = zeros(size(x)); gamma = mu*dt;xADiC(1) = x(1); Q3 = x(1); Q1 = x(1); for i = 2:length(x); % QTFs    dX = x(i) - Q3;     if dX > -0.5*gamma & dX < 1.5*gamma Q3 = x(i);    else Q3 = Q3 + gamma∗(sign(dX)+0.5); end     dX = x(i) - Q1;    if dX > -1.5*gamma & dX < 0.5*gamma Q1 = x(i);    else Q1 = Q1 + gamma∗(sign(dX)-0.5); end % FENCES    alpha_p(i) = Q3 + beta∗(Q3-Q1);     alpha_m(i) = Q1 - beta∗(Q3-Q1);% ADiC output    if x(i)>alpha_p(i) | x(i)<alpha_m(i) xADiC(i) = 0.5*(Q3+Q1);    else xADiC(i) = x(i); end end return

FIG. 54 provides and example of applying a numerical version of an ADiCshown in FIG. 51 to the input signal used in FIG. 15 , illustrating thatits performance is similar to that of the MATLAB function of §2.5.

8.1 Cascaded ADiC Structures

To improve suppression of outliers, two or more ADiCs may be cascaded,as illustrated in FIG. 55 . Since an ADiC depreciates outliers, thefences in a subsequent ADiC may be made “tighter” as they would be lessaffected by the reduced (depreciated) outliers, thus enabling furtheroutlier reduction.

This is illustrated in FIG. 56 . The top panel in the figure shows thesignal of interest affected by outlier noise (the signal x(t) in FIG. 55). While, as shown in the panel second from the top, the “intermediate”fences

α^(′)₊(t)andα^(′)⁻(t)

(e.g. constructed using QTFs) would be relatively robust, they wouldstill be affected by the outliers, and the range

[α^(′)⁻, α^(′)₊]

for the outlier depreciation may be unnecessarily wide. Since only thoseoutliers that extend outside of the range

[α^(′)⁻, α^(′)₊]

would be depreciated, the “intermediate” ADiC output y′(t) would stillcontain outliers that do not protrude from the range formed by

α^(′)₊(t)andα^(′)⁻(t).

This may be seen in the middle panel of FIG. 56 .

Since the outliers in y′(t) are reduced in comparison with those inx(t), the fences α₊(t) and α₋(t) around y′(t) may be made “tighter” asthey would be less affected by the reduced (depreciated) outliers, asmay be seen in the panel second from the bottom in FIG. 56 . This mayenable further outlier reduction, since the outliers that extend outsideof the reduced range [α₋,α₊] would be depreciated, as may be seen in thebottom panel of the figure.

9 ADiC-Based Filtering of Complex-Valued Signals

In a number of applications it may be desirable to perform ADiC-basedfiltering of complex-valued signals. For example, since the power oftransient interference in a quadrature receiver would be shared betweenthe in-phase and the quadrature channels, the complex-valued processing(as opposed to separate processing of the in-phase/quadraturecomponents) may have a potential of significantly improving theefficiency of the ADiC-based interference mitigation [8-10, 32, forexample].

In a complex-valued ADiC with the input z(t) and the DCL ζ(t), outliersmay be identified based on a magnitude of the complex-valued differencesignal, e.g. based on |z(t) - ζ(t)|.

For example, a complex-valued CMTF may be constructed as illustrated inFIG. 57 .

In FIG. 57 , an influence function

ℐ_(α²)(x)

is represented as

ℐ_(α²)(x) = x𝒯_(α²)(x²),

where

𝒯_(α²)(x²)

is a transparency function with the characteristic transparency rangeα². We may require that

𝒯_(α²)(x²)

is effectively (or approximately) unity for x² ≤ α², and that

𝒯_(α²)(x²)

becomes smaller than unity (e.g. decays to zero) as x² increases forx² > α².

As one should be able to see in FIG. 57 , a nonlinear differentialequation relating the input z(t) to the output ζ(t) of a complex-valuedCMTF may be written as

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\zeta = \frac{1}{\tau}I_{\alpha^{2}}\left( {z - \zeta} \right) = \frac{z - \zeta}{\tau}T_{\alpha^{2}}\left( \left| {z - \zeta} \right|^{2} \right),} & \text{­­­(53)}\end{matrix}$

where τ is the CMTF’s time parameter (or time constant).

One skilled in the art will recognize that, according to equation (53),when the magnitude of the difference signal |z(t)-ζ(t)|² is within thetransparency range, |z-ζ|² ≤ α², the complex-valued CMTF would behave asa 1st order linear lowpass filter with the 3 dB corner frequency1/(2πτ), and, for a sufficiently large transparency range, the CMTFwould exhibit nonlinear behavior only intermittently, when the magnitudeof the difference signal extends outside the transparency range.

If the transparency range α²(t) is chosen in such a way that it excludesoutliers of |z(t)-ζ(t)|², then, since the transparency function

𝒯_(α²)(x²)

decreases (e.g. decays to zero) for x² > α², the contribution of suchoutliers to the output ζ(t) would be depreciated.

It may be important to note that outliers would be depreciateddifferentially, that is, based on the magnitude of the difference signal|z(t)-ζ(t)|² and not the input signal z(t).

The degree of depreciation of outliers based on their magnitude woulddepend on how rapidly the transparency function

𝒯_(α²)(x²)

decreases (e.g. decays to zero) for x² > α². For example, as followsfrom equation (53), once the transparency function decays to zero, theoutput ζ(t) would maintain a constant value until the magnitude of|z(t)-ζ(t)|² returns to within non-zero values of the transparencyfunction.

In FIG. 57 , double-line arrows correspond to complex-valued signals,while single-line arrows correspond to real-valued signals.

An example of a numerical algorithm implementing a numerical version ofa complex-valued ADiC with the DCL formed by a complex-valued CMTF maybe given by the following MATLAB function:

function [zADiC,zCMTF,dZsq_A,Q1,Q3,alpha] = ADiC_complex(z,dt,tau,beta,mu)%------------------------------------------------------------------------------Ntau = (1+floor(tau/dt)); A = 1;%------------------------------------------------------------------------------zADiC = zeros(size(z)); zCMTF = zeros(size(z)); dZsq_A = zeros(1,length(z));Q1 = zeros(1,length(z)); Q3 = zeros(1,length(z));alpha = zeros(1,length(z)); gamma = mu*dt;%------------------------------------------------------------------------------zADiC(1) = z(1); zCMTF(1) = z(1); dZsq_A(1) = 0;Q1(1) = 0; Q3(1) = 0; alpha(1) = 0; Balpha = 0;%------------------------------------------------------------------------------for i = 2:length(z);   dZ = z(i)-zCMTF(i-1); dZsq_A(i) = dZ*conj(dZ)/A;%------------------------------------------------------------------------------  dZ_ = dZsq_A(i) - Q3(i-1);   if dZ_ > -0.5*gamma & dZ_ < 1.5*gamma    Q3(i) = dZsq_A(i);   else    Q3(i) = Q3(i-1) + gamma*(sign(dZ_)+0.5);   end  dZ_ = dZsq_A(i) - Q1(i-1);   if dZ_ > -1.5*gamma & dZ_ < 0.5*gamma    Q1(i) = dZsq_A(i);   else    Q1(i) = Q1(i-1) + gamma*(sign(dZ_)-0.5);   end%------------------------------------------------------------------------------% TUKEY’S upper fence   alpha(i) = Q3(i) + beta*(Q3(i)-Q1(i));%------------------------------------------------------------------------------  if dZsq_A(i) > alpha(i)     Balpha = 0;   else     Balpha = dZ;   end  zCMTF(i) = zCMTF(i-1) + Balpha/Ntau;   zADiC(i) = Balpha + zCMTF(i-1);end return

FIG. 58 provides an illustration of using a complex-valued ADiC formitigation of impulsive interference (e.g. OOB interference from adigital communication transmitter) affecting the signal in a quadraturereceiver. The leftmost panels show the in-phase (I) and the quadrature(Q) traces of the baseband QRSK-modulated received signal affected by amixture of Gaussian (e.g. thermal) and impulsive noise, observed at abandwidth significantly wider (e.g. several times or an order ofmagnitude wider) than the bandwidth of the signal of interest. In alinear receiver, this signal would be digitized, filtered with a matchedfilter, and appropriately sampled to obtain the received symbols.However, the power of the impulsive noise in the signal bandwidth issignificant, which results in a noisy, low SNR output and high errorrates. This may be seen from the constellation diagram shown in the topof the rightmost panels.

Since the power of the interference would be shared between the in-phaseand the quadrature channels, we may treat the I and Q traces as acomplex-valued signal z(t) = I(t) + iQ(t), and apply a complex-valuedADiC for mitigation of this interference before downsampling andapplying a matched filter. As one may see from the constellation diagramshown in the bottom of the rightmost panels in FIG. 58 , the ADiC filtersuppresses the impulsive part of the interference affecting the basebandsignal, increasing the SNR and decreasing the BER.

In FIG. 58 , double-line arrows correspond to complex-valued signals.

As illustrated in FIG. 59 , a complex-valued ADiC structure may also begiven the following alternative description. In the figure, double-linearrows correspond to complex-valued signals, while single-line arrowscorrespond to real-valued signals.

First, a complex-valued Differential Clipping Level (DCL) ζ(t) is formedby an analog or digital DCL circuit. Such a DCL may be established as anoutput of a robust (i.e. insensitive to outliers) filter estimating alocal Measure of Central Tendency (MCT) of the complex-valued inputsignal z(t). A complex-valued MCT filter may be formed, for example, bytwo real-valued MCT filters applied separately to the real and theimaginary components of z(t). Another example of a complex-valued MCTfilter would be a complex-valued Median Tracking Filter (MTF) describedin the next paragraph.

Complex-valued Median Tracking Filter - Let us consider the signal ζ(t)related to a complex-valued signal z(t) by the following differentialequation:

$\begin{matrix}{\frac{\text{d}}{\text{d}t}\zeta = \frac{A}{T}{sgn}\left( {z - \zeta} \right) = \mu{sgn}\left( {z - \zeta} \right),} & \text{­­­(54)}\end{matrix}$

where A is a parameter with the same units as |z| and |ζ|, T is aconstant with the units of time, and the signum (sign) function isdefined as sgn(z) = z/|z|. The parameter µ may be called the slew rateparameter. Equation (54) would describe the relation between the inputz(t) and the output ζ(t) of a particular robust filter forcomplex-valued signals, the Median TrackingFilter (MTF).

Then, the difference signal z(t) - ζ(t) is obtained.

Next, a robust range α(t) for the magnitude of the difference signal isdetermined, by a Robust Range Circuit (RRC). Such a range may be, e.g.,a robust upper fence α(t) constructed for |z(t)-ζ(t)| as a linearcombination of the outputs of quantile tracking filters applied to|z(t)-ζ(t)|. Or, as shown in FIG. 59 , such a range may be, e.g., arobust upper fence α²(t) constructed for |z(t)-ζ(t)|² as a linearcombination of the outputs of quantile tracking filters applied to|z(t)-ζ(t)|².

The magnitude of the difference signal and the upper fence are the inputsignals of the depreciator characterized by a transparency function andproducing the output, e.g.,

𝒯_(α)(|z − ζ|)

or

𝒯_(α²)(|z − ζ|²),

used for depreciation of outliers. Specifically, the ADiC output v(t)may be set to be equal to a weighted average of the input signal z(t)and the DCL ζ(t), with the weights given by the depreciator output

𝒯_(α)(|z − ζ|)

or

𝒯_(α²)(|z − ζ|²)

as follows:

$\begin{matrix}{v = \zeta + \left( {z - \zeta} \right)T_{\alpha}\left( \left| {z - \zeta} \right| \right),} & \text{­­­(55)}\end{matrix}$

or, as shown in FIG. 59 ,

$\begin{matrix}{v = \zeta + \left( {z - \zeta} \right)T_{\alpha^{2}}\left( \left| {z - \zeta} \right|^{2} \right).} & \text{­­­(56)}\end{matrix}$

For example, for the transparency function given by a boxcar function,the ADiC output v(t) would be equal to the ADiC input z(t) when thedifference signal is within the range (e.g. α(t) or α²(t)), and it wouldbe equal to the DCL ζ(t) otherwise:

$\begin{matrix}{v = \left\{ \begin{matrix}z & {\text{­­­(57)}\left| {z - \zeta} \right| \leq \alpha} \\\zeta & \text{otherwise}\end{matrix} \right)\mspace{6mu}.} & \end{matrix}$

An example of a numerical algorithm implementing a numerical version ofa complex-valued ADiC with the DCL formed by a complex-valued MTF, aboxcar depreciator, and a robust upper fence α²(t) constructed for|z(t)-ζ(t)|² using QTFs, may be given by the following MATLAB function:

function [zADiC,zMTF,dZsq_A,alpha] = ADiC_MTF_complex(z,dt,mu_MTF,mu_range,beta)%------------------------------------------------------------------------------zADiC = zeros(size(z)); zMTF = zeros(size(z)); dZsq_A = zeros(1,length(z));alpha = zeros(1,length(z)); gamma_MTF = mu_MTF*dt; gamma_range = mu_range*dt;%------------------------------------------------------------------------------zADiC(1) = z(1); zMTF(1) = z(1); dZsq_A(1) = 0; alpha(1) = 0; Q3 = 0; Q1 = 0%------------------------------------------------------------------------------for i = 2:length(z); dZ = z(i)-zMTF(i-1); dZsq_A(i) = dZ*conj(dZ);%------------------------------------------------------------------------------% MEDIAN TRAKING FILTER (MTF) applied to incoming signal    if abs(dZ) < gamma_MTF          zMTF(i) = z(i);     else         zMTF(i) = zMTF(i-1) + gamma_MTF∗(sign(dZ));     end%------------------------------------------------------------------------------% QUARTILE TRACKING FILTERS (QTFs) applied to squared difference signal    dZ_ = dZsq_A(i) - Q3;    if dZ_ > -0.5*gamma_range & dZ_ < 1.5*gamma_range         Q3 = dZsq_A(i);     else         Q3 = Q3 + gamma_range∗(sign(dZ_)+0.5);     end    dZ_ = dZsq_A(i) - Q1;    if dZ_ > -1.5*gamma_range & dZ_ < 0.5*gamma_range         Q1 = dZsq_A(i);     else         Q1 = Q1 + gamma_range∗(sign(dZ_)-0.5);     end%------------------------------------------------------------------------------% TUKEY’S upper fence     alpha(i) = Q3 + beta∗(Q3-Q1);%------------------------------------------------------------------------------% ADiC output    if dZsq_A(i)>alpha(i) zADiC(i) = zMTF(i); else zADiC(i) = z(i); endend return

10 Hidden Outlier Noise and its Mitigation

In addition to ever-present thermal noise, various communication andsensor systems may be affected by interfering signals that originatefrom a multitude of other natural and technogenic (man-made) phenomena.Such interfering signals often have intrinsic temporal and/or amplitudestructures different from the Gaussian structure of the thermal noise.Specifically, interference may be produced by some “countable” or“discrete”, relatively short duration events that are separated byrelatively longer periods of inactivity. Provided that the observationbandwidth is sufficiently large relative to the rate of thesenon-thermal noise generating events, and depending on the noise couplingmechanisms and the system’s filtering properties and propagationconditions, such noise may contain distinct outliers when observed inthe time domain. The presence of different types of such outlier noiseis widely acknowledged in multiple applications, under various generaland application-specific names, most commonly as impulsive, transient,burst, or crackling noise.

While the detrimental effects of EMI are broadly acknowledged in theindustry, its outlier nature often remains indistinct, and itsomnipresence and impact, and thus the potential for its mitigation,remain greatly underappreciated. There may be two fundamental reasonswhy the outlier nature of many technogenic interference sources is oftendismissed as irrelevant. The first one is a simple lack of motivation.As discussed in this disclosure, without using nonlinear filteringtechniques the resulting signal quality would be largely invariant to aparticular time-amplitude makeup of the interfering signal and woulddepend mainly on the total power and the spectral composition of theinterference in the passband of interest. Thus, unless the interferenceresults in obvious, clearly identifiable outliers in the signal’s band,the “hidden” outlier noise would not attract attention. The secondreason is highly elusive nature of outlier noise, and inadequacy oftools used for its consistent observation and/or quantification. Forexample, neither power spectral densities (PSDs) nor their short-timeversions (e.g. spectrograms) allow us to reliably identify outliers, assignals with very distinct temporal and/or amplitude structures may haveidentical spectra. Amplitude distributions (e.g. histograms) are alsohighly ambiguous as an outlier-detection tool. While a super-Gaussian(heavy-tailed) amplitude distribution of a signal does normally indicatepresence of outliers, it does not necessarily reveal presence or absenceof outlier noise in a wideband signal. Indeed, a wide range of powersacross a wideband spectrum would allow a signal containing outlier noiseto have any type of amplitude distribution. More important, theamplitude distribution of a non-Gaussian signal is generally modifiableby linear filtering, and such filtering may often convert the signalfrom sub-Gaussian into super-Gaussian, and vice versa. Thus apparentoutliers in a signal may disappear and reappear due to various filteringeffects, as the signal propagates through media and/or the signalprocessing chain.

10.1 “Outliers” vs. “Outlier Noise”

Even when sufficient excess bandwidth is available for outlier noiseobservation, outlier noise mitigation faces significant challenges whenthe typical amplitude of the noise outliers is not significantly largerthan that of the signal of interest. That would be the case, e.g., ifthe signal of interest itself contains strong outliers, or for largesignal-to-noise ratios (SNRs), especially when combined with high ratesof the noise-generating events. In those scenarios removing outliersfrom the signal+noise mixture may degrade the signal quality instead ofimproving it. This is illustrated in FIG. 60 . The left-hand side of thefigure shows a fragment of a low-frequency signal affected by a widebandnoise containing outliers. However, the amplitudes of the signal and thenoise outliers are such that only one of the outlier noise pulses is anoutlier for the signal+noise mixture. The right-hand side of the figureillustrates that removing only this outlier increases the basebandnoise, instead of decreasing it by the “outlier noise” removal.

10.2 “Excess Band” Observation for In-Band Mitigation

As discussed earlier, a linear filter affects the amplitudes of thesignal of interest, wideband Gaussian noise, and wideband outlier noisedifferently. FIG. 61 illustrates how one may capitalize on thesedifferences to reliably distinguish between “outliers” and “outliernoise”. The left-hand side of the figure shows the same fragment of thelow-frequency signal affected by the wideband noise containing outliersas in the example of FIG. 60 . This signal+noise mixture may be viewedas an output of a wideband front-end filter. When applied to the outputof the front-end filter, a baseband lowpass filter that does notattenuate the low-frequency signal would still significantly reduce theamplitude of the wideband noise. Then the difference between the inputsignal+noise mixture and the output of the baseband filter with zerogroup delay across signal’s band would mainly contain the wideband noisefiltered with highpass filter obtained by spectral inversion of thebaseband filter. This is illustrated in the right-hand side of FIG. 61 ,showing that now the outliers in the difference signal are also thenoise outliers.

Thus detection of outlier noise may be accomplished by an “excess bandfilter” constructed as a cascaded lowpass/highpass (for a basebandsignal of interest), or as a cascaded bandpass/bandstop filter (for apassband signal of interest). This is illustrated in FIG. 62 where, forsimplicity, finite impulse response (FIR) filters are used. Providedthat the “excess band” is sufficiently wide in comparison with the bandof the signal of interest, the impulse response of an excess band filtercontains a distinct outlier component. When convolved with aband-limited signal affected by a wideband outlier noise, such a filterwould suppress the signal of interest while mainly preserving theoutlier structure of the noise. Below we illustrate how such excess bandobservation of outlier noise may enable its efficient in-bandmitigation.

10.3 Complementary ADiC Filter (CAF)

Following the previous discussion in this disclosure, the basic conceptof wideband outlier noise removal while preserving the signal ofinterest and the wideband non-outlier noise may be stated as follows:(i) first, establish a robust range around the signal of interest suchthat this robust range excludes wideband noise outliers; (ii) thenreplace noise outliers with mid-range. When we are not constrained bythe needs for either analog or wideband, high-rate real-time digitalprocessing, in the digital domain these requirements may be satisfied bya Hampel filter or by one of its variants [45]. In a Hampel filter the“mid-range” is calculated as a windowed median of the input, and therange is determined as a scaled absolute deviation about this windowedmedian. However, Hampel filtering may not be performed in the analogdomain, and/or it may become prohibitively expensive in high-ratereal-time digital processing.

As discussed earlier, a robust range [α₋,α₊] that excludes outliers of asignal may also be obtained as a range between Tukey’s fences [48]constructed as linear combinations of the 1st and the 3rd quartiles ofthe signal in a moving time window, or constructed as linearcombinations of other quantiles. In practical analog and/or real-timedigital implementations, approximations for the time-varying quantilevalues may be obtained by means of Quantile Tracking Filters (QTFs)described in Section 3. Linear combinations of QTF outputs may also beused to establish the mid-range that replaces the outlier values. Forexample, the signal values that protrude from the range [α₋,α₊] may bereplaced by (Q_([1]) + wQ_([2]) + Q_([3]))/(w + 2), where w ≥ 0. Thensuch mid-range level may be called a Differential Clipping Level (DCL),and a filter that established the range [α₋,α₊] and replaces outlierswith the DCL may be called an Analog Differential Clipper (ADiC).

As discussed in Section 10.1, for reliable discrimination between“outliers” and “outlier noise” the amplitude of the signal of interestshould be much smaller than a typical amplitude of the noise outliers.Therefore, the best application for an ADiC would be the removal ofoutliers from the “excess band” noise (see Section 10.2), when thesignal of interest is mainly excluded. Then ADiC-based filtering thatmitigates wideband outlier noise while preserving the signal of interestmay be accomplished as described below.

10.3.1 Spectral Inversion by ADiC and “Efecto Cucaracha”

Let us note that applying an ADiC to an impulse response of a highpassand/or bandstop filter containing a distinct outlier would cause the“spectral inversion” of the filter, transforming it into its complement,e.g. a highpass filter into a lowpass, and a bandstop filter into abandpass filter. This is illustrated in FIG. 63 where, for simplicity,FIR filters are used. Thus, as further demonstrated in FIG. 64 , an ADiCapplied to a filtered outlier noise may significantly reshape itsspectrum. Such spectral reshaping by an ADiC may be called a “cockroacheffect”, when reducing the effects of outlier noise in some spectralbands increases its PSD in the bands with previously low outlier noisePSD. We may use this property of an ADiC for removing outlier noisewhile preserving the signal of interest, and for addressing complexinterference scenarios.

10.3.2 Removing Outlier Noise While Preserving Signal of Interest

For example, in FIG. 65 the bandpass filter mainly matches the signal’spassband, and the bandstop filter is its “complement” obtained byspectral inversion of the bandpass filter, so that the sum of theoutputs of the bandpass and the bandstop filters is equal to the inputsignal. The input passband signal of interest affected by a widebandoutlier noise may be seen in the upper left of FIG. 65 . The output ofthe bandpass filter is shown in the upper middle of the figure, wherethe trace marked by “Δ” shows the effect of the outlier noise on thepassband signal. As discussed in Section 10.2, the output of thebandstop filter is mainly the “excess band” noise. After the outliers ofthe excess band noise are mitigated by an ADiC (or another nonlinearfilter mitigating noise outliers), the remaining excess band noise isadded to the output of the bandpass filter. As the result, the combinedoutput (seen in the upper right of the figure) would be equal to theoriginal signal of interest affected by a wideband noise with reducedoutliers. This mitigated outlier noise is shown by the trace marked by“Δ” in the upper right.

FIG. 66 summarizes such “complementary” ADiC-based outlier noise removalfrom band-limited signals. To simplify the mathematical expressions, inFIG. 66 we use zero for the group delay of the linear filters and assumethat the ADiC completely removes the outlier component i(t) from theexcess band. We may call this ADiC-based filtering structure aComplementary ADiC Filter (CAF).

As illustrated in FIG. 67 , a complementary ADiC filter may be given thefollowing description. In a CAF, a “signal filter” (e.g. a lowpass, abandpass, or another filter that mainly matches the desired signal’sband) is applied to the input signal containing the signal of interestand wideband noise and interference. The output of the signal filterprovides a “filtered input signal”. A “complement signal filter” is alsoapplied to the input signal to provide a “complement filtered inputsignal”. The complement filtered input signal is further filtered with anonlinear filter that mitigates outliers in the complement filteredinput signal, providing a “nonlinear filtered complement signal”. Such anonlinear filter may be, for example, an ADiC filter described in thisdisclosure, or a variant of an ADiC filter. Finally, the CAF outputsignal is formed as the sum of the filtered input signal and thenonlinear filtered complement signal.

Note that the sum of the filtered input signal and the complementfiltered input signal would be effectively equal the input signal (e.g.to the time-delayed version of the input signal, based on the groupdelay of the signal filter). Thus the complement filtered input signalmay also be obtained as the difference between a time-delayed inputsignal and the filtered input signal.

11 Explanatory Comments and Discussion

This section of the disclosure provides several comments on thedisclosure given in Sections 1 through 10.

It should be understood that the specific examples in this disclosure,while indicating preferred embodiments of the invention, are presentedfor illustration only. Various changes and modifications within thespirit and scope of the invention should become apparent to thoseskilled in the art from this detailed description. Furthermore, all themathematical expressions, diagrams, and the examples of hardwareimplementations are used only as a descriptive language to convey theinventive ideas clearly, and are not limitative of the claimedinvention.

Further, one skilled in the art will recognize that the variousequalities and/or mathematical functions used in this disclosure areapproximations that are based on some simplifying assumptions and areused to represent quantities with only finite precision. We may use theword “effectively” (as opposed to “precisely”) to emphasize that only afinite order of approximation (in amplitude as well as time and/orfrequency domains) may be expected in hardware implementation.

Ideal vs. “real” blankers - For example, we may say that an output of ablanker characterized by a blanking value is effectively zero when theabsolute value (modulus) of said output is much smaller (e.g. by anorder of magnitude or more) than the blanking range.

In addition to finite precision, a “real” blanker may be characterizedby various other non-idealities. For example, it may exhibit hysteresis,when the blanker’s state depends on its history.

For a “real” blanker, when the value of its input x extends outside ofits blanking range [α₋,α₊], the value of its transparency function woulddecrease to effectively zero value over some finite range of theincrease (decrease) in x. If said range of the increase (decrease) in xis much smaller (e.g. by an order of magnitude or more) than theblanking range, we may consider such a “real” blanker as beingeffectively described by equations (18), (32) and/or (37).

Further, in a “real” blanker the change in the blanker’s output may be“lagging”, due to various delays in a physical circuit, the change inthe input signal. However, when the magnitude of such lagging issufficiently small (e.g. smaller than the inverse bandwidth of the inputsignal), and provided that the absolute value of the blanker outputdecreases to effectively zero value, or restores back to the inputvalue, over a range of change in x much smaller than the blanking range(e.g. by an order of magnitude or more), we may consider such a “real”blanker as being effectively described by equations (18), (32) and/or(37).

11.1 Mitigation of Non-Gaussian (e.g. Outlier) Noise in the Process ofAnalog-to-Digital Conversion: Analog and Digital Approaches

Conceptually, ABAINFs are analog filters that combine bandwidthreduction with mitigation of interference. One may think of non-Gaussianinterference as having some temporal and/or amplitude structure thatdistinguishes it form a purely random Gaussian (e.g. thermal) noise.Such structure may be viewed as some “coupling” among differentfrequencies of a non-Gaussian signal, and may typically require arelatively wide bandwidth to be observed. A linear filter thatsuppresses the frequency components outside of its passband, whilereducing the non-Gaussian signal’s bandwidth, may destroy this coupling,altering the structure of the signal. That may complicate furtheridentification of the non-Gaussian interference and its separation froma Gaussian noise and the signal of interest by nonlinear filters such asABAINFs.

In order to mitigate non-Gaussian interference efficiently, the inputsignal to an ABAINF would need to include the noise and interference ina relatively wide band, much wider (e.g. ten times wider) than thebandwidth of the signal of interest. Thus the best conceptual placementfor an ABAINF may be in the analog part of the signal chain, forexample, ahead of an ADC, or incorporated into the analog loop filter ofa ΔΣ ADC. However, digital ABAINF implementations may offer manyadvantages typically associated with digital processing, including, butnot limited to, simplified development and testing, configurability, andreproducibility.

In addition, as illustrated in §3.3, a means of tracking the range ofthe difference signal that effectively excludes outliers of thedifference signal may be easily incorporated into digital ABAINFimplementations, without a need for separate circuits implementing sucha means.

While real-time finite-difference implementations of the ABAINFsdescribed above would be relatively simple and computationallyinexpensive, their efficient use would still require a digital signalwith a sampling rate much higher (for example, ten times or more higher)than the Nyquist rate of the signal of interest.

Since the magnitude of a noise affecting the signal of interest wouldtypically increase with the increase in the bandwidth, while theamplitude of the signal+noise mixture would need to remain within theADC range, a high-rate sampling may have a perceived disadvantage oflowering the effective ADC resolution with respect to the signal ofinterest, especially for strong noise and/or weak signal of interest,and especially for impulsive noise. However, since the sampling ratewould be much higher (for example, ten times or more higher) than theNyquist rate of the signal of interest, the ABAINF output may be furtherfiltered and downsampled using an appropriate decimation filter (forexample, a polyphase filter) to provide the desired higher-resolutionsignal at lower sampling rate. Such a decimation filter may counteractthe apparent resolution loss, and may further increase the resolution(for example, if the ADC is based on ΔΣ modulators).

Further, a simple (non-differential) “hard” or “soft” clipper may beemployed ahead of an ADC to limit the magnitude of excessively strongoutliers in the input signal.

As discussed earlier, mitigation of non-Gaussian (e.g. outlier) noise inthe process of analog-to-digital conversion may be achieved by deployinganalog ABAINFs (e.g. CMTFs, ADiCs, or CAFs) ahead of the anti-aliasingfilter of an ADC, or by incorporating them into the analog loop filterof a ΔΣ ADC, as illustrated in FIG. 68 , panels (a) and (b),respectively.

Alternatively, as illustrated in panel (b) of FIG. 68 , awider-bandwidth anti-aliasing filter may be employed ahead of an ADC,and an ADC with a respectively higher sampling rate may be employed inthe digital part. A digital ABAINF (e.g. CMTF, ADiC, or CAF) may then beused to reduce non-Gaussian (e.g. impulsive) interference affecting anarrower-band signal of interest. Then the output of the ABAINF may befurther filtered with a digital filter, (optionally) downsampled, andpassed to the subsequent digital signal processing.

Prohibitively low (e.g. 1-bit) amplitude resolution of the output of aΔΣ modulator would not allow direct application of a digital ABAINF.However, since the oversampling rate of a ΔΣ modulator would besignificantly higher (e.g. by two to three orders of magnitude) than theNyquist rate of the signal of interest, a wideband (e.g. with bandwidthapproximately equal to the geometric mean of the nominal signalbandwidth B_(x) and the sampling frequency F_(s)) digital filter may befirst applied to the output of the quantizer to enable ABAINF-basedoutlier filtering, as illustrated in panel (b) of FIG. 69 .

It may be important to note that the output of such a wideband digitalfilter would still contain a significant amount of high-frequencydigitization (quantization) noise. As follows from the discussion in §3,the presence of such noise may significantly simplify using quantiletracking filters as a means of determining the range of the differencesignal that effectively excludes outliers of the difference signal.

The output of the wideband filter may then be filtered by a digitalABAINF (with appropriately chosen time parameter and the blankingrange), followed by a linear lowpass /decimation filter.

11.2 Comments on Δς Modulators

The 1st order ΔΣ modulator shown in panel I of FIG. 1 may be describedas follows. The input D to the flip-flop, or latch, is proportional toan integrated difference between the input signal x(t) of the modulatorand the output Q. The clock input to the flip-flop provides a controlsignal. If the input D to the flip-flop is greater than zero, D > 0, ata definite portion of the clock cycle (such as the rising edge of theclock), then the output Q takes a positive value V_(c), Q = V_(c). If D< 0 at a rising edge of the clock, then the output Q takes a negativevalue -V_(c), Q = -V_(c). At other times, the output Q does not change.It may be assumed that Q is the compliment of Q and Q = -Q.

Without loss of generality, we may require that if D = 0 at a clock’srising edge, the output Q retains its previous value.

One may see in panel I of FIG. 1 that the output Q is a quantizedrepresentation of the input signal, and the flip-flop may be viewed as aquantizer. One may also see that the integrated difference between theinput signal of the modulator and the output Q (the input D to theflip-flop) may be viewed as a particular type of a weighted differencebetween the input and the output signals. One may further see that theoutput Q is indicative of this weighted difference, since the sign ofthe output values (positive or negative) is determined by the sign ofthe weighted difference (the input D to the flip-flop).

One skilled in the art will recognize that the digital quantizer in a ΔΣmodulator may be replaced by its analog “equivalent” (i.e. Schmitttrigger, or comparator with hysteresis).

Also, the quantizer may be realized with an N-level comparator, thus themodulator would have a log₂(N)-bit output. A simple comparator with 2levels would be a 1-bit quantizer; a 3-level quantizer may be called a“1.5-bit” quantizer; a 4-level quantizer would be a 2-bit quantizer; a5-level quantizer would be a “2.5-bit” quantizer.

11.3 Comparators, Discriminators, Clippers, and Limiters

A comparator, or a discriminator, may be typically understood as acircuit or a device that only produces an output when the input exceedsa fixed value.

For example, consider a simple measurement process whereby a signal x(t)is compared to a threshold value D. The ideal measuring device wouldreturn ‘0’ or ‘1’ depending on whether x(t) is larger or smaller than D.The output of such a device may be represented by the Heaviside unitstep function θ (D - x(t)) [30], which is discontinuous at zero. Such adevice may be called an ideal comparator, or an ideal discriminator.

More generally, a discriminator/comparator may be represented by acontinuous discriminator function

ℱ_(α)(x)

with a characteristic width (resolution) α such that

lim_(α → 0)ℱ_(α)(x) = θ(x).

In practice, many different circuits may serve as discriminators, sinceany continuous monotonic function with constant unequal horizontalasymptotes would produce the desired response under appropriate scalingand reflection. For example, the voltage-current characteristic of asubthreshold transconductance amplifier [51, 52] may be described by thehyperbolic tangent function,

${\widetilde{\mathcal{F}}}_{\alpha}(x) = A\,\,\tanh\left( {x/\alpha} \right).$

Note that

$\lim_{\alpha\rightarrow 0}\frac{{\widetilde{F}}_{\alpha}(x) - A}{2A} = \theta(x),$

and thus such an amplifier may serve as a discriminator.

When α « A, a continuous comparator may be called a high-resolutioncomparator.

A particularly simple continuous discriminator function with a “ramp”transition may be defined as

$\begin{matrix}{F_{g,A}(x) = \left\{ \begin{matrix}{gx} & {\text{­­­(58)}g|x| \leq A} \\{A{sgn}(x)} & \text{otherwise}\end{matrix} \right)\mspace{6mu}\mspace{6mu},} & \end{matrix}$

where g may be called the gain of the comparator, and A is thecomparator limit.

Note that a high-gain comparator would be a high-resolution comparator.

The “ramp” comparator described by equation (58) may also be called aclipping amplifier (or simply a “clipper”) with the clipping value A andgain g.

For asymmetrical clipping values α₊ (upper) and α₋ (lower), a clippermay be described by the following clipping function

C_( α⁻)^(α₊)(x):

$\begin{matrix}{C_{\mspace{6mu}\alpha_{-}}^{\alpha_{+}}(x) = \left\{ \begin{matrix}\alpha_{+} & {\text{­­­(59)}x > \alpha_{+}} \\\alpha_{-} & {\text{for}x < \alpha_{-}} \\x & \text{otherwise}\end{matrix} \right)\mspace{6mu}.} & \end{matrix}$

It may be assumed in this disclosure that the outputs of the activecomponents (such as, e.g., the active filters, integrators, and thegain/amplifier stages) may be limited to (or clipped at) certain finiteranges, for example, those determined by the power supplies, and thatthe recovery times from such saturation may be effectively negligible.

11.4 Windowed Measures of Location

In the current disclosure, a Windowed Measure of Location (WML) would bea summary statistics that attempts to describe a set of data in a giventime window by a single value. Most typically, a measure of location maybe understood as a measure of central location, or central tendency. Aweighted mean (often called a weighted average) would be the mosttypically used measure of central tendency, and it may be called aWindowed Measure of Central Tendency (WMCT). When the weights do notdepend on the data values, a WMCT may be considered a linear measure ofcentral tendency.

An example of a (generally) nonlinear measure of central tendency wouldbe the quasi-arithmetic mean or generalized f-mean [53]. Other nonlinearmeasures of central tendency may include such measures as a median or atruncated mean value, or an L-estimator [48, 54, 55].

A measure of location obtained in a moving time window w(t) would be aWindowed Measure of Location (WML). For example, given an input signalx(t), the output χ(t) of a linear lowpass or bandpass filter with theimpulse response w(t), χ(t) = (w∗x)(t), may represent a linear measureof location of the input signal x(t) in a moving time window w(t).

Note that when

∫_(−∞)^(∞)ds w(s) = 1,

w(t) would represent a lowpass filter, and a linear WML in such a timewindow would be a linear WMCT. However, such w(t) that

∫_(−∞)^(∞)ds w(s) = 0

(e.g., an impulse response of a linear bandpass or bandstop filter) mayalso be used to obtain a linear WML for a signal. For example, if thelinear filter has an effectively unity frequency response and aneffectively zero group delay over the bandwidth of a signal of interest,such a filter may be used to obtain a linear WML for the signal ofinterest affected by an interfering signal.

As another example, let us consider the signal χ(t) implicitly given bythe following equation:

$\begin{matrix}{{\int_{- \infty}^{\infty}{\text{d}s\, w\left( {t - s} \right)\text{sign}\left( {\chi(t) - x(s)} \right)}} = w(t) \ast \text{sign}\left( {\chi - x(t)} \right) = 0,} & \text{­­­(60)}\end{matrix}$

where

∫_(−∞)^(∞)ds w(s) = 1.

Such χ(t) would represent a weighted median of the input signal x(t) ina moving time window w(t), and χ(t) would be a robust nonlinear WML(WMCT) of the input signal x(t) in a moving time window w(t).

One skilled in the art will recognize that such nonlinear filters as amedian filter, a CMTF, an NDL, an MTF, or a TTF would representnonlinear WMLs (i.e. WMCTs) of their inputs.

11.5 Mitigation of Non-Impulsive Non-Gaussian Noise

The temporal and/or amplitude structures (and thus the distributions) ofnon-Gaussian signals are generally modifiable by linear filtering, andnon-Gaussian interference may often be converted from sub-Gaussian intosuper-Gaussian, and vice versa, by linear filtering [9, 10, 32, e.g.].Thus the ability of the ADiCs/CMTFs/ABAINFs/CAFs disclosed herein, andΔΣ ADCs with analog nonlinear loop filters, to mitigate impulsive(super-Gaussian) noise may translate into mitigation of non-Gaussiannoise and interference in general, including sub-Gaussian noise (e.g.wind noise at microphones). For example, a linear analog filter may beemployed as an input front end filter of the ADC to increase thepeakedness of the interference, and the ΔΣ ADCs with analog nonlinearloop filter may perform analog-to-digital conversion combined withmitigation of this interference. Subsequently, if needed, a digitalfilter may be employed to compensate for the impact of the front endfilter on the signal of interest.

Alternatively, increasing peakedness of the interference may be achievedby modifying the wideband filter following the ΔΣ modulator andpreceding the ADiC/CAF, as illustrated in panel (b) of FIG. 70 . In thisexample, the function of the wideband filter with the impulse responseg[k] would be to enhance the distinction between the signal of interestand the outlier noise, thus increasing the efficiency of the outliernoise mitigation by the ADiC/CAF.

The response g[k] of the wideband “outlier-enhancing” filter may be suchthat it affects the signal of interest, e.g. g[k] ∗ w[k] ≠ w[k], wherew[k] is the response of the “original” narrow-band “baseband” filter(such as a lowpass or bandpass filter) of the ΔΣ ADC before the additionof the ADiC-based processing (see panel (a) of FIG. 70 ). In such acase, the filter w[k] may be modified by adding the term Δw[k] tocompensate for the impact of the wideband filter on the signal ofinterest. For example, the term Δw[k] may be chosen to satisfy thefollowing condition:

$\begin{matrix}{g\lbrack k\rbrack \ast \left( {w\lbrack k\rbrack + \Delta w\lbrack k\rbrack} \right) \approx w\lbrack k\rbrack.} & \text{­­­(61)}\end{matrix}$

As an example, let as consider mitigation of wideband impulsive noisethat was previously filtered with a 2nd order bandpass filter such thatthe filtered noise may no longer clearly appear impulsive, as may beseen in the upper left panel of FIG. 71 . The cross-hatched areas in therightmost panels of FIG. 71 correspond to the passband PSD of the“ideal” signal of interest (without interference), and the solid linescorrespond to the PSDs of the filtered signal+noise mixtures.

Since the noise contains non-zero power spectral density in the signal’spassband, a linear passband filter applied directly to thesignal+interference mixture (the panels in row II of FIG. 71 ) affectsthe signal and the interference proportionally in its passband, and itdoes not improve the passband SNR.

While the bandpass-filtered impulsive noise shown in row I of FIG. 71may no longer be a distinct outlier noise that would be efficientlymitigated by an ADiC/CAF, filtering this noise by a 1st order highpassfilter with an appropriate time constant may convert this noise into adistinctly outlier (e.g. impulsive) noise, as illustrated in the leftpanel of row III. As shown in row IV of FIG. 71 , such an outlier noisemay be efficiently mitigated by an ADiC/CAF.

From the differential equation for a 1st order highpass filter it wouldfollow that g_(τ) ∗ [w + (1/τ) ∫[dt w] = w, where the asterisk denotesconvolution and where g_(τ)(t) is the impulse response of the 1st orderlinear highpass filter with the corner frequency 1/(2πτ). Thus, tocompensate for the insertion of a 1st order highpass filter before anADiC/CAF, the digital bandpass filter after the ADiC/CAF may be modifiedby adding a term proportional to an antiderivative of the impulseresponse w[k] of the bandpass filter, w[k] → w[k] + Δw[k] = w[k] + (1/τ)∫dt w[k]. FIG. 72 illustrates the impulse and the frequency responses ofw[k] and w[k] + Δw[k] used in the example of FIG. 71 .

The modified passband filter w[k] + Δw[k] applied to the ADiC/CAF’soutput would suppress the remaining interference outside of thepassband, while compensating for the insertion of the 1st order highpassfilter before the ADiC/CAF. This would result in an increased passbandSNR, as illustrated in the panels of row V in FIG. 71 .

As another illustrative example, let as consider ADiC-based mitigationof wideband impulsive noise affecting the baseband signal of interest inthe presence of a strong adjacent-channel interference.

Let us first notice that an impulse response of a bandstop filter may beconstructed by adding an outlier to an impulse response of a bandpassfilter. Therefore, by removing (e.g. by an ADiC) this outlier from theimpulse response of the bandstop filter the bandstop filter would beeffectively converted to a respective bandpass filter. It then wouldfollow that applying an ADiC filter to an impulsive noise filtered witha bandstop filter may effectively convert the bandstop-filteredimpulsive noise into a respective bandpass-filtered impulsive noise, asillustrated by the idealized example of FIG. 73 .

As schematically shown in FIG. 74 , ADiC-based mitigation of widebandimpulsive noise affecting the baseband signal of interest in thepresence of a strong adjacent-channel interference may be performed asfollows.

First, a bandstop filter is applied to the signal+noise+interferencemixture to effectively suppress (or adequately reduce) the adjacentchannel interference. Then the ADiC filtering is applied to the outputof the bandstop filter, mitigating the impulsive noise in the baseband.Finally, a linear baseband filter is applied to the ADiC’s output,suppressing the remaining interference outside of the baseband.

Let us compare the two signal processing chains shown in FIG. 75 , andinspect the examples of the time-domain traces and the PSDs of thesignals at points I, II, III, IV, and V.

The example input signal (point I in FIG. 75 and the panels in row I ofFIG. 76 ) consists of the baseband signal of interest, a mixture of abroadband-filtered AWGN and a broadband impulsive noise, and anadjacent-channel interference with the PSD in its passband much largerthan that of the impulsive noise and that of the baseband PSD of thesignal of interest.

Since the impulsive noise contains non-zero power spectral density inthe signal’s passband, a linear baseband filter applied directly to thesignal+interference mixture (point II in FIG. 75 and the panels in rowII of FIG. 76 ) affects the signal and the interference proportionallyin its passband, and it does not improve the baseband SNR.

As discussed earlier, when a (narrow-band) baseband signal of interestis affected only by a mixture of a broadband Gaussian and a broadbandimpulsive noise, the latter may be efficiently mitigated by an ADiC.However, as illustrated in the upper left panel of FIG. 76 , thepresence of a strong adjacent-channel interference may “obscure” theimpulsive noise, impeding its identification as “outliers” and making adirect use of an ADiC for its mitigation ineffective.

To enable impulsive noise mitigation, one may first suppress theadjacent-channel interference by a linear bandstop filter, thus“revealing” the impulsive noise (point III in FIG. 75 and the panels inrow III of FIG. 76 ) and making its “pulses” identifiable as outliers.

An ADiC applied to the bandstop-filtered signal would thus be enabled tomitigate the impulsive noise, disproportionately reducing its basebandPSD while raising its PSD in the stopband of the bandstop filter byapproximately the respective amount (point IV in FIG. 75 and the panelsin row IV of FIG. 76 ).

A linear baseband filter applied to the ADiC’s output would suppress theremaining interference outside of the baseband, resulting in anincreased baseband SNR (point V in FIG. 75 and the panels in row V ofFIG. 76 ).

11.6 Clarifying Remarks

“ADiC-based filter” should be understood as a filter comprising an ADiCstructure. For example, an ADiC-based filter may consist of a widebandlinear lowpass filter followed by an ADiC or a CAF followed by a linearbandpass filter. As another example, in FIG. 75 the ADiC-based filterconsists of a bandstop filter followed by an ADiC or a CAF followed by alinear lowpass filter.

As another example of an ADiC-based filter, an “ADiC-based decimationfilter” should be understood as a decimation filter comprising an ADiCor a CAF structure. For example, it may consist of a digital ADiC or aCAF followed by a digital decimation filter.

FIG. 77 provides an example of a ΔΣ ADC with an ADIC-based decimationfilter for mitigation of wideband impulsive noise affecting the basebandsignal of interest in the presence of a strong adjacent-channelinterference. In this example, the ADiC-based decimation filter consistsof (i) a digital wideband filter followed by (ii) a digital ADiC/CAFfollowed by (iii) a digital decimation filter.

The wideband filter may, in turn, consist of a several cascaded filters.For example, for mitigation of wideband impulsive noise affecting thebaseband signal of interest in the presence of a strong adjacent-channelinterference, the wideband filter may consist of a wideband lowpassfilter cascaded with a bandstop filter for suppression of theadjacent-channel interference.

While conceptually the best implementation and use of ADiC-based filtersmay be in analog hardware, as discussed in this disclosure, inherentlyhigh (e.g. by two to three orders of magnitude higher than the Nyquistrate for the signal of interest) oversampling rate of a ΔΣ ADC may beused for a real-time, low memory, and computationally inexpensive“effectively analog” digital ADiC-based filtering duringanalog-to-digital conversion. Such numerical ADiC implementations mayoffer many advantages typically associated with digital processing,including simplified development and testing, on-the-flyconfigurability, reproducibility, and the ability to “train” (optimize)the ADiC parameters (e.g., using machine learning approaches). Inaddition, such an approach may simplify ADiC’s integration into thoseexisting systems that use ΔΣ ADCs for analog-to-digital conversion.

For example, FIG. 78 illustrates a direct conversion receiverarchitecture with quadrature baseband ADCs, where the ADiC-basedfiltering (using either complex-valued processing, or separateprocessing of the in-phase and quadrature components) may be performedearly in the digital domain, immediately following ΔΣ modulators (e.g.1-bit ΔΣ modulators). A high sampling rate of ΔΣ modulators would allowthe use of relaxed analog filtering requirements, e.g. much widerantialiasing filter bandwidth. For instance, one may use a low-orderBessel filter with the 3 dB corner frequency that is an order ofmagnitude wider than that of the baseband, to provide a sufficientbandwidth margin along with preserving the shape of the interference’soutliers.

FIG. 79 shows a generic example of a superheterodyne receiverarchitecture with incorporated ADiC-based filtering. In such a receiver,a baseband stage may amplify, filter, and then A/D convert the resultingin-phase and quadrature signals. After the A/D (performed, e.g., by1-bit ΔΣ modulators), digital ADiC-based filtering may be performed toattenuate interferers before digital detection of the bit sequence isperformed. Alternatively, the IF signal may be digitized by a single ADC(e.g. by a ΔΣ ADC) after which additional filtering (including ADiCfiltering), quadrature downconversion to DC, and bit detection areperformed in the digital domain.

One skilled in the art will recognize that a variety of electroniccircuit topologies may be developed and/or used to implement theintended functionality of various ADiC structures.

FIGS. 80, 81, and 82 outline brief examples of idealized algorithmictopologies for several ADiC sub-circuits based on the operationaltransconductance amplifiers (OTAs). Transconductance cells based on themetal-oxide-semiconductor (MOS) technology represent an attractivetechnological platform for implementation of such active nonlinearfilters as ADiCs, and for their incorporation into IC-based signalprocessing systems. ADiCs based on transconductance cells offer simpleand predictable design, easy incorporation into ICs based on thedominant IC technologies, small size, and can be used from the low audiorange to gigahertz applications [56-59].

For example, FIG. 80 provides a conceptual schematic of a sub-circuitfor an OTA-based implementation of a depreciator with the transparencyfunction given by equation (30) and depicted in FIG. 20 .

FIG. 81 provides an example of an OTA-based squaring circuit (e.g. “SQ”circuit in FIG. 59 ) for a complex-valued signal.

FIG. 82 provides an example of a conceptual schematic of a sub-circuitfor an OTA-based implementation of a depreciator with the transparencyfunction depicted in FIGS. 57 and 59 and given by the followingequation:

$\begin{matrix}{T_{\alpha^{2}}\left( \left| {z - \zeta} \right|^{2} \right) = \left\{ \begin{matrix}1 & {\text{­­­(62)}\left| {z - \zeta} \right| \leq \alpha} \\\left( \frac{\alpha}{\left| {z - \zeta} \right|} \right)^{2} & \text{otherwise}\end{matrix} \right).} & \end{matrix}$

One skilled in the art will recognize that various other OTA-basedsub-circuits for different ADiC embodiments (e.g. implementingaddition/subtraction, multiplication/division, absolute value, squareroot, and other linear and/or nonlinear functions) may be implementedusing the approaches and the circuit topologies described, for example,in [60-63].

Note that if the DCLs χ(t) or ζ(t) in FIGS. 44, 51 or 59 are establishedby filtering the signal x(t) or z(t) with a linear filter with theimpulse response w(t) (i.e. as χ(t) = (w ∗ x)(t) or ζ(t) = (w ∗ z)(t)),having an effectively unity frequency response and an effectivelyconstant group delay Δt > 0 over the bandwidth of the signal ofinterest, they may be viewed as DCLs for the delayed signals x(t-Δt) orz(t-Δt). Then the difference signals for x(t-Δt) or z(t-Δt) (i.e.x(t-Δt) - χ(t) or z(t-Δt) - ζ(t)) may be obtained as the respectiveoutputs of a bandstop filter with the impulse response δ(t-Δt) - w(t),where δ(x) is the Dirac δ-function [31].

12 Utilizing Pileup Effect and Intermittently Nonlinear Filtering inSynthesis of Low-SNR and/or Covert and Hard-to-Intercept CommunicationLinks

To meet the undetectability requirement, in a steganographic system thestego signals should be statistically indistinguishable from the coversignals. For physical layer transmissions, it may perhaps be enhanced byrequiring that the payload and the cover have the same bandwidth andspectral content, the same apparent temporal and amplitude structures,and that there are no explicit differences in the spectral and/ortemporal allocations for the cover signals and the payload messages.

For a mixture of such signals, neither linear nor nonlinear filteringalone may separate the signals. Favorably, however, linear filtering maysignificantly, and differently, affect the temporal and amplitudestructure of many natural and the majority of technogenic (man-made)signals. For example, such filtering may often convert the amplitudedistribution of a pulse train from super-Gaussian into apparentlyGaussian and/or sub-Gaussian, and vice versa. On the other hand, anonlinear filter is capable of disproportionately affecting spectraldensities of signals with distinct temporal and/or amplitude structureseven when the signals have the same spectral content. Therefore, aproper synergistic combination of linear and nonlinear filtering may beemployed to effectively separate such “indistinguishable” cover andstego signals.

12.1 Channel Noise as Cover Signal

The very existence of a detectable carrier (cover signal) may be a deadgiveaway for the stego payload. For example, a simple presence of asheet of paper implies the possibility of a message written in invisibleink. Therefore, the best steganography should be “carrier-less,” whenthe payload is covertly embedded into something “ever-present.” In thephysical layer, such “ideal” and unidentifiable cover signal may be thechannel noise. Such noise always includes the ever-present thermal noiseas one of its components, and may also comprise other (in general,non-Gaussian) natural and/or technogenic (man-made) components. Then, ifthe stego payload “pretends” to be Gaussian, and its power is smallenough to be well within the natural variations of the channel noise,any physically available band may be used to carry a virtuallyundetectable covert message.

In this disclosure, we describe an approach to physical-layersteganography where the transmitted low-power stego messages may bestatistically indistinguishable from the Gaussian component of thechannel noise (e.g. the thermal noise) observed in the same spectralband, and thus the channel noise itself may serve as an effective coversignal. We also demonstrate how the apparent spectral and temporalproperties of transmitted additional, higher-power cover signals(including those using the existing communication protocols) may be madeto match those of the low-power stego payload and the Gaussian noise,providing extra layers of obfuscation for both the cover and the stegomessages. We further illustrate how a specific combination of linear andnonlinear filtering may be used for effective separation of the cover,payload, and/or “friendly jamming” signals even when all transmissionshave effectively the same spectral characteristics as well temporal andamplitude structures, and when there are no explicit differences in thespectral and/or temporal allocations for the cover and the stegomessages.

12.2 Mimicking Function of Pileup Effect

A pulse train p(t) may be simply a sum of pulses with the same shape(impulse response) w(t), same or different amplitudes a_(k), anddistinct arrival times t_(k): p(t) = Σ_(k) a_(k)w(t-t_(k)). When thewidth of the pulses in a train becomes greater than the distance betweenthem, the pulses may begin to overlap and interfere with each other.This is illustrated in FIG. 83 : For the same arrival times, the pulsesin the sequence consisting of the narrow pulses w(t) remain separate,while the wider (more “spread out”) pulses g(t) are “piling up on top ofeach other.” In this example, w(t) and g(t) have the same spectralcontent, and thus the power spectral densities (PSDs) of the pulsesequences are identical. However, the “pileup effect” causes thetemporal and amplitude structures of these sequences to be substantiallydifferent. For a random pulse train, when the ratio of the bandwidth andthe pulse arrival rate becomes significantly smaller than thetime-bandwidth product (TBP) of a pulse, the pileup effect may cause theresulting signal to become effectively Gaussian [43, e.g.], making itimpossible to distinguish between the individual pulses.

Indeed, let p̂(t) be an “ideal” pulse train: p̂(t) = Σ_(k) α_(k)δ(t -t_(k)), where δ(x) is the Dirac δ-function [31]. The moving average ofthis ideal train in a boxcar window of width 2T may be represented bythe convolution integral

$\begin{matrix}{\overline{p}(t) = {\int_{- \infty}^{\infty}{\text{d}s\frac{\theta\left( {t + T} \right) - \theta\left( {t - T} \right)}{2T}\hat{p}\left( {t - s} \right)}},} & \text{­­­(63)}\end{matrix}$

where θ(x) is the Heaviside unit step function [30]. At any given timet_(i), the value of p(t_(i)) is proportional to the sum of α_(k) for thepulses that occur within the interval [t_(i)-T, t_(i)+T]. Then, if theamplitudes α_(k) and/or the interarrival times t_(k+1) - t_(k) areindependent and identically distributed (i.i.d.) random variables withfinite mean and variance, it follows from the central limit theorem(CLT) that the distribution of p̅(t_(i)) approaches Gaussian for asufficiently large interval [-T, T].

If we replace the boxcar weighting function in (63) with an arbitrarymoving window w(t), then (63) becomes a weighted moving average

$\begin{matrix}{p(t) = {\int_{- \infty}^{\infty}{\text{d}s\, w(t)\hat{p}\left( {t - s} \right)}} = \left( {\hat{p} \ast w} \right)(t){\sum\limits_{k}{a_{k}w\left( {t - t_{k}} \right)}}\mspace{6mu},} & \text{­­­(64)}\end{matrix}$

which is a “real” pulse train with the impulse response w(t). If w(t) isnormalized so that

∫_(−∞)^(∞)ds w(s) = 1,

w(t) is an averaging (i.e. lowpass) filter. Then, if w(t) has both thebandwidth and the time-bandwidth product (TBP) similar to that of theboxcar pulse of width 2T, the distribution of p(t_(i)) would be similarto that of p̅(t_(i)) (e.g. Gaussian for a sufficiently large T).

12.2.1 TBP of Filter in Context of Pileup Effect

There are various ways to define the “time duration” and the “bandwidth”of a pulse. This may lead to a significant ambiguity in the definitionsof the TBPs, especially for filters with complicated temporal structuresand/or frequency responses. However, in the context of a mimickingfunction of the pileup effect, our main concern is the change in the TBPthat occurs only due to the change in the temporal structure of afilter, without the respective change in its spectral content. For asingle pulse w(t), its peak-to-average power ratio (PAPR) may beexpressed as

$\begin{matrix}{\text{PAPR}_{w} = \frac{\text{max}\left( {w^{2}(t)} \right)}{\frac{1}{T_{2} - T_{1}}{\int_{T_{1}}^{T_{2}}{\text{d}t\, w^{2}(t)}}}\mspace{6mu},} & \text{­­­(65)}\end{matrix}$

where the interval [T₁, T₂] includes the effective time support of w(t).Then for filters with the same spectral content and the impulseresponses w(t) and g(t), the ratio of their TBPs may be expressed as thereciprocal of the ratio of their PAPRs,

$\begin{matrix}{\frac{\text{TBP}_{g}}{\text{TBP}_{w}} = \frac{\max\left( {w^{2}(t)} \right)}{\max\left( {g^{2}(t)} \right)} = \frac{\text{PAPR}_{w}}{\text{PAPR}_{g}},} & \text{­­­(66)}\end{matrix}$

where the PAPRs are calculated over a sufficiently long time intervalthat includes the effective time support of both filters.

Note that from (66) it follows that, among all possible pulses with thesame spectral content, the one with the smallest TBP would contain adominating large-magnitude peak. Hence any reasonable definition of afinite TBP for a particular filter with a given frequency response mayallow us to obtain comparable numerical values for the TBPs of all otherfilters with the same frequency response, regardless of their temporalstructures. For example, defining the “time duration” of the pulsesg₁(t) and g₂(t) shown in FIG. 84 may be challenging. On the other hand,a “reasonable” definition of the duration of the root-raised-cosinepulse w(t) may be given as 2 T_(s), where T_(s) is the the reciprocal ofthe symbol-rate parameter of the pulse. Then defining the bandwidth bythe 3 dB corner frequency (i.e. ΔB = (2 T_(s))⁻¹) leads to TBP_(w) = 1.

Given a “seed” pulse w(t), perhaps the easiest way to construct a pulseg(t) with the same spectral content but a different TBP is to filterw(t) with an all-pass filter, for example, a linear or nonlinear chirpwith a flat frequency response. Then the convolutions of w(t) and g(t)with their respective matched filters (i.e. their “combined” impulseresponses) would be automatically identical. For example, the pulsesg₁(t) and g₂(t) shown in FIG. 84 are obtained by convolving theroot-raised-cosine pulse w(t) with two different nonlinear chirps. Whilew(t), g₁(t), and g₂(t) have significantly different TBPs, theirconvolutions with the respective matched filters produce the sameraised-cosine pulse (w∗w)(t).

We would like to mention in passing that the same approach may be usedto construct multidimensional pairs of matched filters with identicalspectral characteristics but significantly different time and/or spatialsupports. Such filters, for example, may be spatial 2D (g_(i)(x,y))and/or spatio-temporal 3D (g_(i)(x,y,t)) filters for image and videoprocessing. This is illustrated in FIG. 85 for 2D filters.

12.2.2 Convolution With Large-TBP Filter as Gaussian Mimic Function

FIG. 86 illustrates how the pileup effect may be used to obscure (e.g.to mimic as Gaussian or sub-Gaussian) a large-PAPR (super-Gaussian)transmitted signal, while fully recovering its distinct temporal andamplitude structure in the receiver. In this example, convolution of thepulse train with a large-TBP filter in the transmitter “hides” itsoriginal structure, and the pulses with larger TBPs perform this moreeffectively. This may be seen in FIG. 86 from both the time-domaintraces and the normal probability plots shown in the lower left corner.For a sufficiently large TBP, the distribution of the filtered pulsetrain becomes effectively Gaussian, making it impossible to distinguishbetween the individual pulses.

The filters g_(i)(t) in FIG. 86 are obtained by filtering theroot-raised-cosine pulse w(t) with different all-pass filters, and thusthey have the same frequency responses. While w(t) and g_(i)(t) havesignificantly different TBPs, their convolutions with the respectivematched filters produce the same raised-cosine pulse (w∗w)(t). Hence, inthe receiver, filtering with a respective matched filter effectivelyrestores the train’s original temporal and amplitude structure.

12.3 Pulse Trains for Low-SNR Communications

For sufficiently low pulse rate R (e.g. below half of the bandwidth forTBP = 1), the PAPR of a pulse train is inversely proportional to R, andthe magnitude of the pulses in a train of a given power may be madearbitrarily large by reducing the pulse rate. Thus a pulse trainconsisting of pulses with a small TBP may be effectively used forlow-SNR communications, when the Shannon’s upper limit on the channelcapacity [44] is itself below the bandwidth.

For the most effective use of the pileup effect for conversion of ahigh-PAPR pulse train with a distinct, super-Gaussian temporal andamplitude structure into an effectively Gaussian signal, by filteringthe train with a large-TBP filter, the pulse train needs to berandomized. This may be accomplished by randomizing the amplitude of thepulses in the train, their arrival times, or both. The ways in which thepulse train is randomized affect the ways in which the information maybe encoded and retrieved. For example, if the timing structure of thepulse train is known, synchronous pulse detection may be used.Otherwise, one may need to employ an asynchronous pulse detection (e.g.pulse counting). This, in turn, affects the capacity of the channel.

12.3.1 Pulse Counting vs. Synchronous Pulse Detection

Let us consider a pulse train consisting of pulses with the bandwidth ΔBand a small TBP, so that a single large-magnitude peak in a pulsedominates, and assume that the arrival rate

ℛ

of the pulses is sufficiently small so that pileup is negligible

$\left( {\text{e}\text{.g}\text{.}R \ll \frac{1}{2}{{\Delta B}/\text{TBP}}} \right).$

When the arrival time of a pulse with the peak amplitude A > 0 is known,the probability of detecting this pulse as positive in the presence ofGaussian noise with zero mean and the variance

σ_(n)²

may be expressed as

$\frac{1}{2}\text{erfc}\left( \frac{- A}{\sigma_{\text{n}}\sqrt{2}} \right).$

Then the pulses with the amplitude

$A > \sigma_{\text{n}}\sqrt{2}\text{erfc}^{- 1}\left( {2\varepsilon} \right)$

would have a pulse identification error rate smaller than ε. Forexample, ε ≲ 1.3×10⁻³ for A ≳ 3σ_(n), and ε ≲ 3.2×10⁻⁵ for A ≳ 4σ_(n).

In pulse counting, a pulse is detected when it crosses a certainthreshold. A false positive detection occurs when such crossing isentirely due to noise, and a false negative detection happens when apulse affected by the noise fails to cross the threshold. For a positivethreshold α₊ > 0, the false negative rate would be smaller than ε if theamplitude of a pulse is

$A > \alpha_{+} + \sigma_{\text{n}}\sqrt{2}\text{erfc}^{- 1}\left( {2\varepsilon} \right).$

As shown in [46, 47], for a filtered noise with zero mean and thevariance

σ_(n)²,

its rate of up-crossing the threshold α₊ > 0 may be expressed as

$R_{\mspace{6mu}\text{max}}\exp\left( {- \frac{1}{2}\left( {\alpha_{+}/\sigma_{\text{n}}} \right)^{2}} \right),$

where the saturation rate

ℛ_(max)

is determined entirely by the filter’s frequency response. Then, for theaverage pulse arrival rate

ℛ,

the threshold value needs to be

$\alpha_{+} > \sigma_{\text{n}}\left\lbrack {- 2\ln\left( {{\varepsilon R}/R_{\mspace{6mu}\text{max}}} \right)} \right\rbrack^{\frac{1}{2}}$

in order to keep the false positive rate below ε. For example, for

$\mathcal{R}/\mathcal{R}_{\max} = 1/10,\,\,\alpha_{+}\underset{˜}{>}4.3\sigma_{n}$

for ε = 10⁻³, and α₊ ≳ 4.8σ_(n) for ε = 10⁻⁴. Note that for an ideal“brick wall” lowpass filter with the bandwidth ΔB the saturation rate

$R_{\mspace{6mu}\text{max}} = {{\Delta B}/\sqrt{3}}\mspace{6mu}\lbrack 46\rbrack.$

Hence, for example, for a root-raised-cosine or a raised-cosine filter

$R_{\mspace{6mu}\text{max}} \approx \left( {2T_{\text{s}}\sqrt{3}} \right)^{- 1},$

where T_(s) is the reciprocal of the symbol-rate parameter of thefilter.

For a pulse rate

ℛ

that is sufficiently smaller than

$R_{\mspace{6mu}\text{0}} = \frac{1}{2}{{\Delta B}/\text{TBP}},$

the PAPR of a train of equal-magnitude pulses is inversely proportionalto

ℛ.

This is illustrated in the left panel of FIG. 87 for a pulse trainconsisting of root-raised-cosine pulses. Then, for a givensignal-to-noise ratio (SNR) of a pulse train affected by additiveGaussian noise, and for a given error rate constraint ε, the pulse rateneeds to be sufficiently small to ensure the pulse detection with theerror rate below ε. This is illustrated in the right panel of FIG. 87 ,for both pulse counting and synchronous pulse detection, for 10⁻² ≤ ε ≤10⁻³ and a pulse train consisting of root-raised-cosine pulses. Forexample, as shown in this panel, for the SNR equal to -10 dB the upperrate limits for 10⁻² ≤ ε ≤ 10⁻³ are approximately (2.8 - 4.1) × 10⁻³ ΔBfor pulse counting, and (2.5 - 4.5) × 10⁻² ΔB for synchronous pulsedetection, where ΔB is the bandwidth of the signal. For comparison, theShannon upper limit on channel capacity [44] is shown by the dashedline.

While the rate limit for pulse counting is approximately an order ofmagnitude lower than for synchronous pulse detection, pulse countingdoes not rely on any α priori knowledge of pulse arrival times, and maybe used as a backbone method for pulse detection. Thus it is used in allsubsequent examples of this disclosure. In practice, both pulse countingand synchronous pulse detection may be used in combination. For example,given a constraint on the total power of the pulse train, counting ofrelatively rare, higher-amplidude pulses may be used to establish thetiming patterns for synchronization, and synchronous detection ofsmaller, more frequent pulses may be used for a higher data rate.

12.4 Intermittently Nonlinear Filtering (INF) for Outlier Mitigation andPulse Counting

In general, a nonlinear filter is capable of disproportionatelyaffecting spectral densities of signals with distinct temporal and/oramplitude structures even when these signals have the same spectralcontent. In particular, the separation of a large-PAPR pulse train and asmall-PAPR signal may be viewed as either (i) mitigation of impulsivenoise affecting the small-PAPR signal, or (ii) extraction of impulsivesignal from the small-PAPR background. In the examples below, a specifictype of Intermittently Nonlinear Filters (INF) is used to accomplisheither or both tasks. While various INF configurations, their differentuses, and the approaches to their analog and/or digital implementationsare described previously in this disclosure (e.g. under such names asABAINF, CMTF, ADiC, or CAF), FIG. 88 illustrates their basic concept. Inan INF, the upper and the lower fences establish a robust range thatexcludes high-amplitude pulses while effectively containing thesmall-PAPR component. The prime INF output simply contains the inputsignal in which the outliers (i.e. the pulses that protrude from therange) are replaced with mid-range values. This constitutes mitigationof impulsive noise affecting the small-PAPR signal. The auxiliary INFoutput is the difference between its input and the prime output. This isakin to extraction of impulsive signal from the small-PAPR background(or “pulse counting”).

12.4.1 Robust Range/Fencing in INF

For an INF to be effective in separation of small-PAPR and impulsivesignals regardless of their relative powers, its range needs to berobust (insensitive) to the pulse train. Favorably, for a mixture of asmall-PAPR signal with bandwidth ΔB, and a pulse train with the samebandwidth and the rate sufficiently below

ℛ₀,

when the pileup effect is insignificant, the value of the interquartilerange (IQR) of the mixture is insensitive to the power of the pulsetrain. This is illustrated in FIG. 89 for a pulse train affected byadditive Gaussian noise. Thus robust upper (α₊) and lower (α₋) fencesfor INF may be constructed as linear combinations of the 1st (Q_([1]))and the 3rd (Q_([3])) quartiles of the signal (Tukey’s fences [48])obtained in a moving time window:

$\begin{matrix}{\left\lbrack {\alpha_{-},\alpha_{+}} \right\rbrack = \left\lbrack {Q_{\lbrack 1\rbrack} - \beta\left( {Q_{\lbrack 3\rbrack} - Q_{\lbrack 1\rbrack}} \right),Q_{\lbrack 3\rbrack} + \beta\left( {Q_{\lbrack 3\rbrack} - Q_{\lbrack 1\rbrack}} \right)} \right\rbrack,} & \text{­­­(67)}\end{matrix}$

where α₊, α₋, Q_([1]), and Q_([3]) are time-varying quantities, and β isa scaling parameter of order unity. When an INF is used for pulsecounting in the presence of additive Gaussian noise, the particularvalue of β should be chosen based on the constraint on the relative rateε of false positive detections. Then, as follows from the discussion inSection 12.3.1,

$\begin{matrix}{\beta \approx 1.05 \times \sqrt{\ln\left( \frac{R_{\mspace{6mu}\max}}{\varepsilon R} \right)} - \frac{1}{2}.} & \text{­­­(68)}\end{matrix}$

For example, for R/R_(max) = ⅒, β ≈ 2.7 for ε = 10⁻³, and β ≈ 3.1 for ε= 10⁻⁴.

12.4.2 Quantile Tracking Filters (QTFs) for Robust Fencing

As a practical matter, Quantile Tracking Filters (QTFs) describedearlier in this disclosure are an appealing choice for such robustfencing in INF, as QTFs are analog filters suitable for widebandreal-time processing of continuous-time signals and are easilyimplemented in analog circuitry. Further, their numerical computationsare

𝒪(1)

per output value in both time and storage, which also enables theirhigh-rate digital implementations in real time.

In brief, the signal Q_(q)(t) that is related to the given input x(t) bythe equation

$\begin{matrix}{\frac{\text{d}}{\text{d}t}Q_{q} = \mu\left\lbrack {\lim\limits_{\varepsilon\rightarrow 0}S_{\varepsilon}\left( {x - Q_{q}} \right) + 2q - 1} \right\rbrack\mspace{6mu},} & \text{­­­(69)}\end{matrix}$

where µ is the rate parameter and 0 < q < 1 is the quantile parameter,may be used to approximate (“track”) the q-th quantile of x(t) for thepurpose of establishing a robustrange [α₋, α₊,]. In (69), the comparatorfunction S_(ε)(x) may be any continuous function such that S_(ε)(x) =sgn(x) for |x| >> ε, and S_(ε)(x) changes monotonically from “-1” to “1”so that most of this change occurs over the range [-ε, ε]. For acontinuous stationary signal x(t) with a constant mean and a positiveIQR, the outputs Q_([1])(t) and Q_([3])(t) of QTFs with a sufficientlysmall rate parameter µ would approximate the 1st and the 3rd quartiles,respectively, of the signal obtained in a moving boxcar time window withthe width ΔT of order 2 × IQR/µ >> 〈ƒ〉⁻¹, where 〈ƒ〉is the averagecrossing rate of x(t) with the 1st and the 3rd quartiles of x(t).Consequently, as illustrated in FIG. 90 , the overall behavior of theQTF fencing for a stationary constant-mean signal with a given IQR wouldbe similar to the fencing with the “exact” quartile filters in a movingboxcar window [θ(t) - θ(t-ΔT)] /ΔT, where ΔT = 2× IQR/µ and µ is the QTFrate parameter. However, for a sampling rate F_(s), numericalcomputations of an “exact” quartile require

𝒪(F_(s)ΔTlog (F_(s)ΔT))

per output value in time, and

𝒪(F_(s)ΔT)

in storage, becoming prohibitively expensive for high-rate real-timeprocessing.

12.5 Illustrative Examples

Let us now provide several particular illustrations of utilizing thepileup effect and synergistic combinations of linear and intermittentlynonlinear filtering for synthesis of covert and hard-to-interceptcommunication links.

12.5.1 Message Sent by Pulse Train Pretending to Be Thermal Noise

FIG. 91 depicts the basic concept of the first example. The transmittedlow-power payload signals are statistically indistinguishable from theGaussian component of the channel noise (e.g. the thermal noise)observed in the same spectral band, and therefore the channel noiseitself serves as a sole cover signal. Further, FIG. 92 provides adetailed particular illustration for the basic concept highlighted inFIG. 91 . Here, the message is encoded in a pulse train by both thepolarity of the pulses and their interarrival times. (We only show theasynchronous detection (pulse counting) performed by obtaining theauxiliary output of an INF. If the rules for the interarrival times areknown, synchronous pulse detection can also be used.) To make thisexample more realistic, in the transmitter and the receiver we use bothdigital finite impulse response (FIR) as well as analog (hardware)infinite impulse response (IIR) filters, and include into considerationthe respective digital-to-analog (D/A) and analog-to-digital (A/D)conversions. For example, in an underwater acoustic communication systemw₁(t) may represent the response of the speaker in the transmitter, andw₂(t) — the response of the hydrophone in the receiver.

The channel noise used in the simulation is additive white Gaussiannoise (AWGN), and its power is chosen to lead to the -10 dB SNR in thepassband of the receiver. Note that the noise may also contain, inaddition to Gaussian, a strong outlier component. For example, inunderwater acoustic communications it may contain strong impulsive noiseproduced by snapping shrimp [1-3]. In this case, an additional INF maybe deployed before applying the filter g₁₁(t) in the receiver (e.g. atpoint N), to mitigate this noise component and to increase the apparentSNR.

12.5.2 Further Obscuring Low-SNR Payloads

For a stego pulse train with a given rate, further increasing the powerof the channel noise (say, by 10 dB) may make the pulse trainundetectable. For example, when the pulse rate is higher than theShannon limit for the given SNR, neither synchronous nor asynchronousdetection would be possible (see Section 12.3.1). However, such increasein the channel noise power may be accomplished by an additional pulsetrain, simply disguised as Gaussian. Then an INF in the receiver, incombination with the respective “de-mimicking” filter, may effectivelyremove this additional noise, enabling the detection of the low-powerpayload. In addition, the higher-power pulse train may itself carry alower-security (or decoy) message, and/or the timing information thatenables synchronous pulse detection in the stego pulse train. Recoveringthis information from the “extra cover” signal would still requireknowledge of the respective mimic filter used by the transmitter. Thisconcept is schematically illustrated in FIG. 93 , and FIGS. 94 and 95provide its detailed walk-through example. Note that even after theeffective removal of the higher-SNR pulse train from the mixture (by thefirst INF), the stego message is still Gaussian, and still hidden behindthe channel noise (and the remainder of the decoy/timing/“extra cover”signal). Thus its recovery still requires knowledge of the second mimicfilter (g₁₂(t)) used by the transmitter.

Filter properties. The main properties of the filters used in thisexample are listed in the lower right panel of FIG. 95 , and the impulseresponses of these filters and their convolutions are illustrated inFIG. 96 . In construction of these filters, we used the approach brieflyoutlined in Section 12.2.1. In general, given the smallest-TBP filterg₀(t) with a particular frequency response, one may construct a greatvariety of filters g_(i)(t) with the same frequency response but muchlarger TBPs (e.g., orders of magnitude larger). These filters may beconstructed in such a way that (i) their combined matched responses areequal to each other, g_(i)(t) _(*) g_(i)(-t) = g_(j)(t) _(*) g_(j)(-t)for any i and j, and have a small TPB, but (ii) the convolutions of anyg_(i)(t) with itself (for i ≠ 0), or with gj(±t) (for i ≠ j) have largeTBPs. For a given “seed” pulse g₀(t), perhaps the easiest way toconstruct a pulse g_(i)(t) with a different TBP is to filter g₀(t) withan all-pass filter, for example, a linear or nonlinear chirp with a flatfrequency response.

12.5.3 Friendly In-Band Jamming

In our third example, the main message is transmitted using one of theexisting communication protocols, but its temporal and amplitudestructure is obscured by employing a large-TBP filter in thetransmitter, e.g., made to be effectively Gaussian. This alone providesa certain level of security, since the intersymbol interference maybecome excessively large and the signal may not be recovered in thereceiver without the knowledge of the mimic filter. In addition, ajamming pulse train, disguised as Gaussian by another (and different)large-TBP filter, is added to the main signal. This jamming signal haseffectively the same spectral content as the main signal, and its poweris sufficiently large (e.g. similar to the main signal) so that the mainsignal is unrecoverable even if the first mimic filter is known. In thereceiver, the jamming pulse train is removed from the mixture (andrecovered, if it itself contains information), enabling the subsequentrecovery of the main message. This concept is schematically illustratedin FIG. 97 .

OFDM PAPR reduction. In addition to improved security, applying alarge-TBP filter to the main signal reduces PAPR of large-crest-factorsignals such as those in orthogonal frequency-division multiplexing(OFDM), as illustrated in FIG. 98 . Here, the simulated OFDM signals aregenerated without restrictions of the proportion of “ones” and “zeros”in a symbol, and thus they have the maximum achievable PAPRs (i.e. 2N,where N is the number of carriers).

Walk-through example. In FIG. 99 , the main signal is a high-PAPR OFDMsignal, and the jamming signal is a high-PAPR impulse train with thespectral content in an effectively the same band (see the frequencyresponses of the filters in the lower left panel of FIG. 99 ). After thefiltering with large-TBP filters g(t) and h(t), respectively, both theOFDM and the jamming signals become effectively Gaussian, and so doestheir mixture that is being transmitted and received (see the black linein the normal probability plots shown in the lower middle panel of FIG.99 ). In this example, the channel noise is assumed to be relativelysmall and is not shown. However, applying a filter matched for h(t) inthe receiver restores the high-PAPR structure of the jamming signal (seethe respective line in the normal probability plots), while the OFDMcomponent remains Gaussian. Subsequently, the INF accomplishes both themitigation of the jamming pulse train affecting the OFDM component andthe extraction of the jamming signal. Applying the filter g̃(t) to theprime INF output effectively restores the original high-PAPR OFDMsignal. If desired, the jamming pulse train is restored by applying thefilter W₂(t) to the auxiliary INF output.

12.6 Pulse Trains for Low-SNR Communications

Let us consider a pulse train consisting of pulses with the bandwidth ΔBand a small TBP, so that a single large-magnitude peak in a pulsedominates, and assume that the arrival rate R of the pulses issufficiently small so that pileup is negligible

$\left( {\text{e}\text{.g}\text{.}R \ll R_{\mspace{6mu} 0} = \frac{1}{2}{{\Delta B}/\text{TBP}}} \right).$

When the arrival time of a pulse with the peak magnitude |A| is known,the probability of correctly detecting the polarity of this pulse in thepresence of additive white Gaussian noise (AWGN) with zero mean and

σ_(n)²

variance may be expressed, using the complementary error function, as

$\frac{1}{2}\text{erfc}\left( \frac{- |A|}{\sigma_{\text{n}}\sqrt{2}} \right).$

Then the pulses with the magnitude

$|A| > \sigma_{\text{n}}\sqrt{2}\,\text{erfc}^{- 1}\left( {2\varepsilon} \right)$

would have a pulse identification error rate smaller than ε. Forexample, ε ≲ 1.3×10⁻³ for |A| ≳ 3σ_(n), and ε ≲ 3.2×10⁻⁵ for |A| ≳4σ_(n).

The pulse rate in a digitally sampled train with regular (periodic)arrival times is R = F_(s/)N_(p), where F_(s) is the sampling frequencyand N_(p) is the number of samples between two adjacent pulses in thetrain. For R that is sufficiently smaller than R₀, the PAPR of a trainof equal-magnitude pulses with regular arrival times is an increasingfunction of the number of samples between two adjacent pulses N_(p), andwould be proportional to N_(p):

$\begin{matrix}{\text{PAPR} = \text{PAPR}\left( N_{\text{p}} \right) \propto N_{\text{p}}\text{for large}N_{\text{p}}\,.} & \text{­­­(70)}\end{matrix}$

For example, for raised-cosine (RC) pulses R₀ ≈ (4T_(S))⁻¹, where T_(s)is the symbol-period, and a “large N_(p)” would mean N_(p) >> T_(s)F_(s)= N_(s), where N_(s) is the number of samples per symbol-period. Asillustrated in FIG. 100 , PAPR(N_(P)) ≈ 1.143 N_(p)/N_(s) forN_(p)/N_(s) >> 1 for RC pulses with roll-off factor β=½.

From the lower limit on the magnitude of a pulse for a given uncoded biterror rate (BER),

$\begin{matrix}{|A| = \sigma_{\text{n}}\sqrt{\text{SNR} \times \text{PAPR}} > \sigma_{\text{n}}\sqrt{2}\,\text{erfc}^{- 1}\left( {2 \times \text{BER}} \right),} & \text{­­­(71)}\end{matrix}$

we may then obtain the lower limit on the SNR for a given pulse rate:

$\begin{matrix}{\text{SNR}\left( {N_{\text{p}};\text{BER}} \right) > \frac{2\left\lbrack {\text{erfc}^{- 1}\left( {2 \times \text{BER}} \right)} \right\rbrack^{2}}{\text{PAPR}\left( N_{\text{p}} \right)} \propto N_{\text{p}}^{- 1},} & \text{­­­(72)}\end{matrix}$

or

$\begin{matrix}{\text{SNR}\left( {N_{\text{p}};\text{BER}} \right) \gtrsim 1.75\left\lbrack {\text{erfc}^{- 1}\left( {2 \times \text{BER}} \right)} \right\rbrack^{2}\frac{N_{\text{s}}}{N_{\text{p}}}} & \text{­­­(73)}\end{matrix}$

for N_(s)/N_(p) << 1 and RC pulses with β= ½. For example, SNR(N_(p);10⁻³) ≳ 9.6/PAPR(N_(p)) ≈ 8.4 N_(s/)N_(p), and SNR(N_(p); 10⁻⁵) ≳18.2/PAPR(N_(p)) ≈ 15.9 N_(s)/N_(p).

FIG. 101 illustrates the SNR limits for different BER as functions ofsamples between pulses for RC pulses with β = ½ and N_(s) = 2. Forexample, for the pulses separated by 128 symbol-periods, BER ≲ 10-3 isachieved for SNR ≳ -12 dB. For comparison, the AWGN Shannon capacitylimit [44] for the bandwidth W = F_(s/)(2N_(s)), which is the nominalbandwidth of the respective RRC filter, is also shown.

12.6.1 Means for Synchronous Detection

To enable synchronous detection for a train x[k] with the pulsesseparated by N_(p) samples, the following modulo power averaging (MPA)function may be constructed as an exponentially decaying average of theinstantaneous signal power x²[k] in a window of size N_(p) +1:

$\begin{matrix}\begin{matrix}{\overline{\text{p}}\left\lbrack {i;k_{j - 1},M} \right\rbrack = \frac{M - 1}{M}\overline{\text{p}}\left\lbrack {i;k_{j - 2},M} \right\rbrack} \\{+ \frac{1}{M}{\sum\limits_{k}{x^{2}\lbrack k\rbrack{〚{k \geq k_{j - 1} - N_{\text{p}}}〛}{〚{k \leq k_{j - 1}}〛}\left. 〚{i =} \right)}}} \\{\left( {{mod}\left( {k,N_{\text{p}}} \right)}〛 \right.,}\end{matrix} & \text{­­­(74)}\end{matrix}$

where k_(j) if the sample index of the j-th pulse, and M>1. In (74), thedouble square brackets denote the Iverson bracket [64]

$\begin{matrix}{{〚P〛} = \left\{ \begin{matrix}1 & {\text{­­­(75)}P\text{is true}} \\0 & \text{otherwise}\end{matrix} \right)\mspace{6mu},} & \end{matrix}$

where P is a statement that may be true or false. Thus the windowk_(j-1)- N_(p) ≤ k ≤ k_(j-1) includes two transmitted pulses, k_(j-2)and k_(j-1), and the index i in p[i; k_(j-1), M] takes the values i = 0,..., N_(p)-1. Note that using exponentially decaying average in (74)would roughly correspond to averaging N =2 M-1 of such windows. Theexponentially decaying average, however, has the advantage of lowercomputational and memory burden, especially for large M, and fasteradaptability to dynamically changing conditions.

For a sufficiently large M the peak in p[i; k-₁, M] corresponding to thepulses of the pulse train would dominate. Therefore, the index k_(j) forsampling of the j-th pulse may be obtained as

$\begin{matrix}{k_{j} = i_{\max} + jN_{\text{p}},} & \text{­­­(76)}\end{matrix}$

where i_(max) is given implicitly by

$\begin{matrix}{\overline{\text{p}}\left\lbrack {i_{\max};k_{j - 1},M} \right\rbrack = \max\left( {\overline{\text{p}}\left\lbrack {i;k_{j - 1},M} \right\rbrack} \right).} & \text{­­­(77)}\end{matrix}$

FIG. 102 illustrates this synchronization procedure. The MPA functionshown in the right-hand side of the figure is computed according to(74). To emphasize the robustness of this synchronization technique evenwhen the bit error rates are very high, the SNR is chosen to berespectively low (SNR = -20 dB, BER ≈ ⅓).

For the link shown in FIG. 107 , FIG. 103 compares the calculated(dashed lines) and the simulated (markers connected by solid lines)BERs, for the “ideal” synchronization (dots), and for thesynchronization with the MPA function described above. The AWGN noise isadded at the receiver input, and the SNR is calculated at the output ofthe matched filter in the receiver. One can see that for M = 2(asterisks) the errors in synchronization are relatively high, whichincreases the overall BER, but the MPA function with M = 8 (crosses)provides reliable yet still fast synchronization. The BERs and therespective SNRs in FIG. 103 are presented for the pulse repetition ratesindicated by the vertical dashed lines in FIG. 101 .

When a pulse train is used for communications rather than, say, radarapplications, reliable synchronization may only need to be achievablefor relatively low BER, e.g. BER ≲ ⅒. Then the following modulomagnitude averaging (MMA) function may replace the MPA function in thesynchronization procedure, in order to reduce the computational burdenby avoiding squaring operations:

$\begin{matrix}{\overline{\text{a}}\left\lbrack {i;k_{j - 1},M} \right\rbrack\begin{array}{l}{= \mspace{6mu}\frac{M - 1}{M}\overline{\text{a}}\left\lbrack {i;k_{j - 2},M} \right\rbrack} \\{+ \mspace{6mu}\mspace{6mu}\frac{1}{M}{\sum\limits_{k}{|x|\lbrack k\rbrack{〚{k > k_{j - 1} - N_{\text{p}}}〛}{〚{k \leq k_{j - 1}}〛}{〚{i = {mod}\left( {k,N_{\text{p}}} \right)}〛}.}}}\end{array}} & \text{­­­(78)}\end{matrix}$

Note that the window k_(j-1) -N_(p) < k ≤ k_(j-1) in (78) includes onlythe (j-1)-th transmitted pulse, instead of two pulses used in (74).

When a correct synchronization has already been obtained, and the maximaare “locked” at the correct i_(max) values, both the MPA and the MMAfunctions would adequately maintain the position of their maxima.However, an offset in the synchronization (e.g. by n points)significantly more unfavorably affects the margin between the extrema ati_(max) and i_(max)+n in the MMA function, compared with the MPAfunction. Thus the “extra point” may cause the “failure to synchronize”even at a relatively high SNR, and it should be removed from thecalculation of the MMA function. Then, as illustrated in FIG. 104 , forBER ≲ ⅒ synchronization with the MMA function a[i; k_(j-1), M] would beeffectively equivalent to synchronization with the MPA function p[i;k_(j-1), M]. When reliable synchronization for larger BERs is desired(e.g. in timing and ranging applications), then the MPA given by (74)should be used.

One skilled in the art will recognize that various other moduloaveraging functions may be used as means for synchronous detection.

For example, the coincidence pulse detection (CPD) function cpd[k] takesthe value “1” if at k there is a local maximum of x[k] that is above

a_(k)⁺,

or a local minimum that is below α _(k) . Othewise, cpd[k] is zero:

$\begin{matrix}\begin{matrix}{\text{cpd}\lbrack k\rbrack = {〚{x_{k} > a_{k}^{+}}〛}{〚{x_{k} > x_{k - 1}}〛}{〚{x_{k} \geq x_{k + 1}}〛}} \\{+ {〚{x_{k} < a_{k}^{-}}〛}{〚{x_{k} < x_{k - 1}}〛}{〚{x_{k} \leq x_{k + 1}}〛}.}\end{matrix} & \text{­­­(79)}\end{matrix}$

If the transmitted pulse rate is ℛ = F_(s)/N_(p) << F_(s), where F_(s)is the sample rate, then N_(p) is the number of samples between twoadjacent pulses in the train. To enable synchronous pulse detection inthe receiver, the following modulo count averaging (MCA) function may beconstructed by the “modulo accumulation” of the values of the pulsedetection function in a window of size MN_(p)+1 that includes M+1transmitted pulses:

$\begin{matrix}\begin{array}{l}{\overline{\text{c}}\left\lbrack {i;k_{j - 1},M} \right\rbrack =} \\{{\sum\limits_{k}{\text{cpd}\lbrack k\rbrack{〚{k \geq k_{j - 1} - MN_{\text{p}}}〛}{〚{k \leq k_{j - 1}}〛}{〚{i = {mod}\left( {k,N_{p}} \right)}〛}}},}\end{array} & \text{­­­(80)}\end{matrix}$

where k_(j) if the sample index of the j-th pulse. Note that in (80) theindex i takes the values i = 0, ..., N_(p)-1. To reduce computations andmemory requirements when M >> 1, the MCA function can also be calculatedas an exponential moving average:

$\begin{matrix}\begin{matrix}{\overline{\text{c}}\left\lbrack {i;k_{j - 1},M} \right\rbrack = \frac{M - 1}{M}\overline{\text{a}}\left\lbrack {i;k_{j - 2},M} \right\rbrack} \\{+ \frac{1}{M}{\sum\limits_{k}{\text{cpd}|x|\lbrack k\rbrack{〚{k \geq k_{j - 1} - N_{\text{p}}}〛}{〚{k \leq k_{j - 1}}〛}{〚{i = {mod}\left( {k,N_{\text{p}}} \right)}〛}}}.}\end{matrix} & \text{­­­(81)}\end{matrix}$

12.7 Summary and Additional Comments

The main results of Section 12 so far may be summarized as follows:

1 - Pileup effect may be used for modifying the temporal and amplitudestructure of various non-Gaussian signals, and, in many cases, formaking them appear as effectively Gaussian. For example, a highlysuper-Gaussian pulse train consisting of pulses with random amplitudesand/or interarrival times may be converted into an effectively Gaussianor sub-Gaussian by a convolution with a filter having a sufficientlylarge time-bandwidth product (TBP). Such “mimicking” of a pulse train asGaussian noise may be achieved without modifying the spectral content ofthe train.

2 - Given the smallest-TBP filter g₀(t) with a particular frequencyresponse, one may construct a great variety of filters g_(i)(t) with thesame frequency response but much larger TBPs (e.g., orders of magnitudelarger). These filters may be constructed in such a way that (i) theircombined matched responses are equal to each other,g_(i)(t)_(*)g_(i)(-t) = g_(j)(t)_(*)g_(j)(-t) for any i and j, and havea small TPB, but (ii) the convolutions any of g_(i)(t) with itself (fori ≠ 0), or with g_(j)(±t) (for i ≠ j) have large TBPs.

There are multiple ways to construct pulses with identical frequencyresponses yet significantly different TBPs. For example, for a given“seed” pulse g₀(t), one of the ways to construct a pulse g_(i)(t) with adifferent TBP may be to filter g₀(t) with an all-pass filter. Such afilter, e.g., may be a linear or nonlinear chirp with a flat frequencyresponse.

As another example, given a “seed” small-TBP pulse with finite (FIR) orinfinite (IIR) impulse response w(t), a large-TBP pulse with the samespectral content may be “grown” from w(t) by applying a sequence of IIRallpass filters. Then an FIR filter for pulse shaping in the transmittermay be obtained by (i) placing w(t) at t = 0, (ii) “spreading” it withan IIR allpass filter, (iii) truncating the pulse when it sufficientlydecays to zero, and (iv) time-inverting the resulting waveform. Thenapplying the same IIR allpass filter in the receiver to this waveformwould produce the matched filter to the original seed pulse, w(-t). Inthe illustration of FIG. 105 , the transmitter waveform is composed as a“piled-up” sum of thus constructed large-TBP pulses, scaled andtime-shifted. In the receiver, an IIR allpass filter recovers theunderlying high-PAPR pulse train. The seed w(t) used in thisillustration is an FIR root-raised-cosine (RRC) pulse symmetrical aroundt = 0, and thus (w_(*)w)(t) is a raised-cosine (RC) pulse. RC pulses areperhaps not the best choice for shaping the pulse trains forcommunications, since their TBP is only about unity, and pulse shapingwith Gaussian or Bessel filters (with TBP ≈ 2ln(2)/π ≈ 0.44) may providea better alternative. However, FIR RC pulses with a given roll-offfactor β have well-defined bandwidth and convinient numerical valuesassociated with their symbol-rate.

3 - Matched filter pairs with similar properties (i.e. identicalspectral characteristics but significantly different time and/or spatialsupports) may also be constructed for multidimensional filters, forexample spatial 2D (g_(i)(x, y)) and/or spatio-temporal 3D (g_(i)(x, y,t)) filters for image and/or video processing.

4 - For sufficiently low pulse rate ℛ (e.g. below half of the bandwidthfor TBP = 1), the PAPR of a pulse train would be inversely proportionalto ℛ, and the magnitude of the pulses in a train of a given power may bemade arbitrarily large by reducing the pulse rate. Thus a pulse trainconsisting of pulses with a small TBP may be effectively used forlow-SNR communications, when the Shannon’s upper limit on the channelcapacity is itself below the bandwidth. For example, if the timingstructure of the pulse train is known, synchronous pulse detection maybe used. Then, in the presence of additive Gaussian noise and for atrain consisting of equal-magnitude pulses with unit TBP, the pulseswith the arrival rates in the 25% to 50% range of the Shannon’s limitfor a given SNR may be detected with the raw error rate in the range10⁻² ≤ ε ≤ 10⁻³. Using proper modulation of the pulse train (e.g. interms of the pulse amplitudes and their interarrival times), and errorcorrection coding, the data rate capacity of a pulse train may bebrought closer to the Shannon’s limit.

5 - When the pulse arrival times are unknown (e.g. the interarrivaltimes are random), the asynchronous pulse detection (pulse counting) maybe used. In pulse counting, a pulse is detected when it crosses acertain threshold, and this threshold needs to be sufficiently high toensure a low rate of false positive detections. Therefore, to ensure acomparable to the synchronous pulse detection error rate, for pulsecounting the pulse arrival rate needs to be reduced by about an order ofmagnitude, down to a few percent of the respective Shannon’s rate. Forexample, to 56-82 kHz for a 20 MHz channel at -10 dB SNR and 10⁻² ≤ ε ≤10⁻³, as compared to 500-900 kHz at the same SNR for synchronousdetection. In practice, both pulse counting and synchronous pulsedetection may be used in combination. For example, given a constraint onthe total power of the pulse train, counting of relatively rare,higher-amplidude pulses may be used to establish the timing patterns forsynchronization, and synchronous detection of smaller, more frequentpulses may be used for a higher data rate.

6 - When each of two or more (say, N) pulse trains consists ofidentically shaped pulses, then, in general, their mixture may not beeffectively separated back into the individual pulse trains. (That is,unless interference among the trains is negligible and a sufficientinformation about the pulse arrival times in the individual pulse trainsis available.) However, before the mixing, one may filter each of theindividual pulse trains with “its own” large-TBP g_(i)(t), i = 1, ...,N, so that the mixture becomes an effectively Gaussian signal due topileup effect. One may then apply to the mixture the filter g_(i)(-t)such that the pulse g_(i)(t) _(*) g_(i)(-t) has the smallest TBP for thegiven spectral content, but the convolutions g_(j)(t) _(*)g_(i)(-t) forj ≠ i would still have sufficiently large TBPs so that the mixture ofthe remaining N-1 pulse trains remains a Gaussian signal. This filteredmixture may then be viewed as (i) a large-PAPR pulse train affected byadditive Gaussian noise, or as (ii) an effectively Gaussian signalaffected by impulsive noise.

7 - In general, a nonlinear filter is capable of disproportionatelyaffecting spectral densities of signals with distinct temporal and/oramplitude structures even when these signals have the same spectralcontent. In particular, the separation of a large-PAPR pulse train and asmall-PAPR signal may be viewed as either (i) mitigation of impulsivenoise affecting the small-PAPR signal, or (ii) extraction of impulsivesignal from the small-PAPR background. In this disclosure, a specifictype of Intermittently Nonlinear Filters (INF) may be used to accomplisheither or both tasks. In such filtering, the upper and the lower fencesestablish a robust range that excludes high-amplitude pulses whileeffectively containing the small-PAPR component. The prime output of anINF would contain the input signal in which the outliers (i.e. thepulses that protrude from the range) are replaced with mid-range values.This would constitute mitigation of impulsive noise affecting thesmall-PAPR signal. The auxiliary INF output would be the differencebetween its input and the prime output. This would be akin to extractionof impulsive signal from the small-PAPR background (or “pulsecounting”).

8 - For an INF to be effective in separation of small-PAPR and impulsivesignals regardless of their relative powers, its range needs to berobust (insensitive) to the pulse train. Favorably, for a mixture of asmall-PAPR signal with bandwidth ΔB, and a pulse train with the samebandwidth and the rate sufficiently below

$R_{0} = {{\frac{1}{2}\Delta B}/\text{TBP}},$

when the pileup effect is insignificant, the value of the interquartilerange (IQR) of the mixture would be insensitive to the power of thepulse train. Thus robust upper and lower fences for INF may beconstructed as linear combinations of the 1st and the 3rd quartiles ofthe signal (Tukey’s fences) obtained in a moving time window. As apractical matter, Quantile Tracking Filters (QTFs) are an appealingchoice for such robust fencing in INF, as QTFs are analog filterssuitable for wideband real-time processing of continuous-time signalsand are easily implemented in analog circuitry. Further, their numericalcomputations are

𝒪(1)

per output value in both time and storage, which also enables theirhigh-rate digital implementations in real time.

9 - The very existence of a detectable carrier (cover signal) may be adead giveaway for the stego payload. For example, a simple presence of asheet of paper implies the possibility of a message written in invisibleink. Therefore, the best steganography should be “carrier-less,” whenthe payload is covertly embedded into something “ever-present.” In thephysical layer, such “ideal” and unidentifiable cover signal would bethe channel noise. Such noise would always include the ever-presentthermal noise as one of its components, and may also comprise other (ingeneral, non-Gaussian) natural and/or technogenic (man-made) components.Then, if the stego payload “pretends” to be Gaussian, and its power issmall enough to be well within the natural variations of the channelnoise, any physically available band may be used to carry a virtuallyundetectable covert message.

10 - Further, Section 12 provides several detailed examples of applyingthe above concepts to synthesis of covert and hard-to-interceptcommunication links. These examples include (i) using the channel noiseas a sole cover signal for a low-power payload, (ii) additionalobfuscation of a low-power messages by strong decoy and/orauxiliary/timing signals, and (iii) “friendly” jamming by a signal withthe same spectral content as the main signal that uses a standardprotocol. All these examples rely on pileup effect for PAPR control, andon combinations of INF and linear filtering for effective separation ofstatistically indistinguishable, same-spectral-band cover and payloadsignals.

11 - Note that when the channel noise itself contains an outliercomponent, an INF deployed early in the receiver chain may mitigate suchoutlier noise, increasing the overall SNR and the throughput capacity ofall channels in the receiver.

One skilled in the art will recognize that the approach described inthis disclosure allows for many practical variations, ranging fromsimple and easily implementable to more elaborate, highly securemulti-level configurations.

12.7.1 Additional Comments on Section 12

PAPR and K_(dBG) as measures of peakedness: The measure of peakedness ofa signal used in Section 12 is PAPR. For deterministic waveforms, PAPRmay be a reliable and consistent measure. However, PAPR may not beappropriate for quantifying peakedness of random signals, especially forlarge data sets, since sample maximum power is the least robuststatistic and is maximally sensitive to outliers, By itself, a PAPRvalue does not quantify the frequency of occurrence of such outliers.For example, a sample of a random Gaussian signal may contain alarge-magnitude outlier, leading to a deceptively large PAPR value.Therefore, instead of using a PAPR value directly, a probability thatPAPR exceeds a certain threshold PAPR₀ is often used to describepeakedness of a random signal. Such probability is a function of PAPR₀and not a statistic (a single value).

It may be more appropriate to measure the peakedness of a signal (e.g.of a pulse train) in terms of its kurtosis in relation to the kurtosisof the Gaussian (aka normal) distribution, as described in Section 4.3.2(see equation (35)), using the units of “decibels relative to Gaussian”(dBG). According to this measure, a Gaussian distribution would havezero dBG peakedness, while sub-Gaussian and super-Gaussian distributionswould have negative and positive dBG peakedness, respectively. In termsof the amplitude distribution of a signal, a higher peakedness comparedto a Gaussian distribution (super-Gaussian) normally translates into“heavier tails” than those of a Gaussian distribution. In the timedomain, high peakedness implies more frequent occurrence of outliers,that is, an impulsive signal.

FIG. 106 provides a comparative illustration of PAPR and K_(dBG) asmeasures of peakedness for pulse trains. As may be seen in the figure,K_(dBG) is less sensitive to outliers than PAPR, and is also moreappropriate for quantifying peakedness of a pulse train at high pulsearrival rates. Thus, when increasing and/or decreasing of signal’speakedness is referenced, the use of a measure such as K_(dBG) may beassumed for quantification of peakedness.

For example, “low peakedness” may be understood as K_(dBG) ≲ 3dBG, and“high peakedness” may be understood as K_(dBG) ≳6 dBG.

Modulation, demodulation, and other functions performed in transmitterand receiver: The examples in Section 12 show in detailed onlyprocessing/filtering of the baseband signals, whereas in a practicalimplementations of transmitters and/or receivers the signal processingchain may include various additional stages and components (e.g. antennacircuits, amplifiers, modulators and demodulators, mixers, various DSPmodules, A/D and D/A converters, oscillators, clocks, input and outputdevices, etc.). For example, some of such components are indicated inFIGS. 1, 2, 21, 22, 26, 27, 32, 36, 38, 40, 42, 51 , FIGS. 68-70 , FIG.75 , and FIGS. 77-79 of this disclosure. Such components areconventional features in various communication apparatus, and theirdetailed illustration is not essential for a proper understanding of thecurrent invention.

In particular, a modulator is a device that performs modulation. Atypical aim of modulation (e.g. digital modulation) is to transfer aband-limited signal (e.g. signal carrying analog or digital bit streaminformation) over a bandpass analog communication channel, for example,over a limited radio frequency band. A demodulator (or “detector”) is adevice that performs demodulation, the inverse of modulation. A modem(from modulator/demodulator) may perform both operations. Modulatorsand/or demodulators are conventional features of various communicationtransmitters and/or receivers, and their detailed illustration is notessential for a proper understanding of the current invention.

FIG. 107 provides an illustration of an apparatus for low-SNR and/orcovert communications, capable of conveying information from atransmitter to a receiver. In this illustration, the physical signalsent from the transmitter (TX) to the receiver (RX) is generated bymodulating a carrier (the signal represented by the signal produced bythe local oscillator LO) with the low-peakedness band-limited modulatingsignal that contains the intended information. For simplicity, only onecomponent of such modulating signal is shown, and the modulating signalmay comprise a plurality of different components. Also, in this figure(as well as for the simulation results discussed above and illustratedin FIG. 103 ) the analog amplitude modulation is used, while other typesof modulation, analog as well as digital, may be used for generation ofthe physical communication signal.

In FIG. 107 , the information sent from TX to RX is encoded in thelow-peakedness band-limited modulating signal. In this illustration, theinformation is contained in the “designed” pulse sequence (train) ∑_(i)A_(i) [[t = t_(i)]] shown in the lower left corner of the figure, andmay be encoded in this designed pulse train by (i) the polarities of thepulses (i.e. sign(A_(i))), (ii) their magnitudes (i.e. |A_(i)|), and/or(iii) the time intervals between different pulses (i.e. t_(i)-t_(j)). Toobtain the low-peakedness band-limited modulating signal, large-TBPpulse shaping is applied to the designed pulse train.

The physical signal is received by RX and the demodulated (e.g.baseband) signal is produced. As shown in FIG. 107 , this demodulatedsignal would comprise a component that is effectively proportional tothe low-peakedness band-limited modulating signal in TX. Further, thiscomponent of the demodulated signal is converted into a high-peakednessband-limited pulse train that corresponds (e.g. in terms of thepolarities, magnitudes, and/or time intervals among the pulses) to thedesigned pulse train, and thus contains the information encoded in thelatter. In FIG. 107 , such conversion is accomplished by applying alarge-TBP filter that is a matched filter to the filter used for pulseshaping in TX.

The intended information may then be extracted from the RX pulse train,by synchronous and/or asynchronous means. For example, the pulses in theRX pulse train may be sampled at their peaks (e.g. at t = t[k] when theCPD function given by (79) returns “1”, cpd[k] = 1), thus providing theinformation about the pulses’ polarities, magnitudes, and/or arrivaltimes.

While in the examples of Section 12 the filtering operations are denotedby the asterisk as convolutions, it may not imply that there are anyspecific requirements imposed on the implementation of such filtering.For example, in FIGS. 86, 92, 94 and 99 it is shown that the transmittedwaveform is generated through filtering (e.g. convolution) of asmall-TBP pulse train with an impulse response of a large-TBP pulse.Instead, such transmitted waveform may be constructed as a simple sum ofscaled and time-shifted/delayed large-TBP pulses (e.g., as ∑_(i) A_(i)g(t - t_(i)), where A_(i) and t_(i) are the amplitude and the arrivaltime of i-th pulse), and no explicit multiplication operations may benecessary for generation of such waveform. Then, for example, instead ofusing a numerical convolution with a high-order FIR filter, in thereceiver the filtering may be performed by several cascaded low-orderIIR allpass filters and a single low-order FIR filter, which may besignificantly less computationally intensive. This is illustrated inFIG. 105 , where the transmitter waveform is constructed as a sum ofscaled and time-shifted/delayed large-TBP pulses. In the receiver, anIIR allpass filter is used to recover the small-TBP pulse train.

As should be seen from FIG. 105 , the low-peakedness signal shown in theleft and the high-peakedness pulse train shown in the right both encodethe same information about the quantities of the “designed” pulse train∑i A_(i) [[t = t_(i)]]: (i) the polarities of the pulses (i.e.sign(A_(i))), (ii) their magnitudes (i.e. |A_(i)|), and (iii) the timeintervals between different pulses (i.e. t_(i)-t_(j)). However, thisinformation may be difficult to recover directly by sampling thelow-peakedness signal, since these quantities are mutually coupledthrough the convolution with a large-TBP filter (pileup). On the otherhand, the polarities, magnitudes, and/or the time intervals among thepulses of the designed pulse train may be easily obtained from themeasuring said quantities in the high-peakedness pulse train shown inthe right of FIG. 105 .

13 Communicating Over Longer Distances at Lower Power and EnergyDissipation

Another object of the present invention is data communications and, inparticular, communicating over longer distances at lower power andenergy dissipation. For example, in low-power wide-area networks(LPWANs), various trade-offs among the bandwidth, data rates, and energyper bit may have different effects on the quality of service underdifferent propagation conditions (e.g. fading and multipath), Dopplerspreads, interference scenarios, multi-user requirements, and designconstraints. Such compromises, and the manner in which they areimplemented, may further affect other technical aspects, such assystem’s computational complexity and power efficiency. At the sametime, this difference in trade-offs may also add to the technicalflexibility in addressing a broader range of communicationsapplications, both static and mobile. In the communications method andapparatus of the present invention the control of the quality of serviceis performed through the change in the spectral efficiency (i.e., thedata rate at a given bandwidth), and/or through changing the energy perbit as an additional trade-off parameter.

13.1 Aggregate Spread Pulse Modulation (ASPM)

For data communications, the present invention introduces the AggregateSpread Pulse Modulation (ASPM), where the information is encoded in theamplitudes A_(j) and/or the “arrival times” k_(j) of the pulses in adigital designed “pulse train” x̂[k] with only relatively small fractionof samples having non-zero values:

$\begin{matrix}{\hat{x}\lbrack k\rbrack = {\sum\limits_{j}{〚{k = k_{j}}〛}}A_{j},} & \text{­­­(82)}\end{matrix}$

where k_(j) is the sample index of the j-th pulse, A_(j) is itsamplitude, and the double square brackets denote the Iverson bracket[64]. The average “pulse rate” ƒ_(p) in such a train is ƒ_(p) =F_(s/)N_(p), where F_(s) is the sample rate, and N_(p) = 〈k_(j) -kj₋₁〉 is the average interpulse interval. Note that for N_(p) >> 1 thepulse rate is much smaller than the Nyquist rate. Also note that forN_(p) >> 1 this train has a large PAPR even when |A_(j)| = const, and isgenerally unsuitable for use as a modulating signal.

However, the designed pulse train x̂[k] given by (82) may be “re-shaped”by linear filtering:

$\begin{matrix}{x\lbrack k\rbrack = \left( {\hat{x} \ast \hat{g}} \right)\lbrack k\rbrack = {\sum\limits_{j}{A_{j}\hat{g}\left\lbrack {k - k_{j}} \right\rbrack}},} & \text{­­­(83)}\end{matrix}$

where ĝ[k] is the impulse response of the filter and the asteriskdenotes convolution. The filter ĝ[k] may be, for example, a lowpassfilter with a given bandwidth B. If the filter ĝ[k] has a sufficientlylarge TBP [65, 66], most of the samples in the reshaped train x[k] willhave non-zero values, and x[k] will have a much smaller PAPR than thedesigned sequence x̂[k]. Such low-PAPR signal may then be used formodulating a carrier. If the combination of the amplitude A_(j) and thearrival time k_(j) of a pulse provides M distinct “states,” each pulsemay encode log₂ M bits, the raw bit rate ƒ_(b) in such a train is ƒ_(b)= ƒ_(p) log₂ M, and such signaling may be referred to as “M-ary.” WhenB >> ƒ_(b) = (F_(s/)N_(p)) log₂ M, it would result in a low-rate messageencoded in a wideband waveform.

For example, for the arrival times in (82) one may use

$\begin{matrix}{k_{j} = jN_{p} + \Delta k\left\lbrack m_{j} \right\rbrack,} & \text{­­­(84)}\end{matrix}$

where Δk is a positive integer, 0 ≤ Δk[m_(j)] < N_(p), and Δk[m] ≠ Δk[l]for m ≠ l. Then for m_(j) = 1,2,...,M and A_(j) = const the pulse traingiven by (82) encodes log₂ M bits per pulse. We may refer to such M-aryencoding with A_(j) = const as “unipolar.” Another bit may be added byusing A_(j) = (-1)^(aj), where α_(j) is either “0” or “1,” and we mayrefer to such signaling as “bipolar.” Then for bipolar M-ary signalingequation (82) may be rewritten as

$\begin{matrix}{\hat{x}\lbrack k\rbrack = {\sum\limits_{j}{{〚{k = jN_{\text{p}} + \Delta k\left\lbrack m_{j} \right\rbrack}〛}\left( {- 1} \right)^{aj}}},} & \text{­­­(85)}\end{matrix}$

where m_(j) = 1, 2, ..., M/2 and α_(j) is either “0” or “1.”

As discussed earlier in this disclosure, for a given designed pulsesequence x̂[k] the spectral, temporal and amplitude structures of thereshaped train x[k] would be determined by the choice of ĝ[k]. Inparticular, it may be desirable to select a filter ĝ[k] that minimizesthe PAPR of x[k]. Note that if the time duration of ĝ[k] extends overmultiple interpulse intervals, the instantaneous amplitudes and/orphases [67] of the resulting waveform are no longer representative ofindividual pulses. Instead, they are a “piled-up” aggregate of thecontributions from multiple “stretched” pulses.

The key property of the large-TBP pulse shaping filter (PSF) ĝ[k] isthat its autocorrelation function (ACF), i.e., the convolution of ĝ[k]with its matched filter g[k] = ĝ[-k], has a much smaller TBP, inparticular, sufficiently smaller than the ratio B/ƒ_(p). Then, afterdemodulation and A/D conversion in the receiver, the encoded binarysequence may be recovered by filtering with g[k] and sampling theresulting pulse train at k = jN_(p)+Δk[m] (i.e., using g[k] as adecimation filter).

A good choice for the PSF would be a pulse that combines a small TBP ofits ACF (e.g., close to that of a Gaussian pulse) with ACF’s compactfrequency support. An example would be a raised-cosine (RC) filter [68,e.g] with unity roll-off factor. The minimum required (Nyquist) samplerate for such a filter will be double its (baseband) physical bandwidthB, and the sample rate F_(s) may be expressed as F_(s) = 2N_(s)B, whereN_(s) ≥ 1 is the oversampling factor. To minimize the power consumption,the memory usage, and the computational complexity of the digitalprocessing, it may be beneficial to keep the sample rate in thetransceivers as low as possible, i.e., to use N_(s) = 1. Through therest of the disclosure, we will typically assume sampling with theNyquist rate F_(s) = 2B.

Since for a given designed pulse sequence x̂[k] the temporal andamplitude structures of the reshaped train x[k] are determined by thePSF ĝ[k], these structures may be substantially different even for thepulse shaping filters with the same ACF. As discussed eariler, one mayconstruct a great variety of large-TBP pulse shaping filters ĝi[k] withthe same small-TBP ACF w[k], so that (ĝ_(i *) g_(i))[k] = w[k] for anyi, while the convolutions of any g_(i)(t) with g_(j)(t) for i ≠ j(cross-correlations) have large TBPs. Further, this property may alsoeffectively hold for the PSFs ĥ_(i)[k] such that ĥ_(i)[k] approximatesthe discrete Hilbert transform of ĝ_(i)[k], i.e., ĥ_(i)[k] = H{ĝ_(i)[k]}[30, 69]. Therefore, using various PSFs combinations we may designdifferent coherent and noncoherent modulation schemes with emphasis onparticular spectral and/or temporal properties of the modulated signal.

13.1.1 Binary (“One Bit Per Pulse”) Encoding

For example, we may construct single-sideband, constant-envelopecoherent and noncoherent ASPM configurations that use the “equidistant”designed train

$\begin{matrix}{\hat{x}\lbrack k\rbrack = {\sum\limits_{j}{{〚{k = jN_{\text{p}}}〛}\left( {- 1} \right)^{bj}}}} & \text{­­­(86)}\end{matrix}$

to encode the binary sequence (b₁b₂... bj...). The raw bit rate ƒ_(b) insuch a train is ƒ_(b) = F_(s/)N_(p), where F_(s) is the sample rate andN_(p) is the number of samples between pulses.

The main challenge for using coherent ASPM for LPWANs may be the needfor carrier recovery. At low SNR (e.g. < -20 dB), combined withsignificant delay and Doppler spreads in multipath environments, suchrecovery may perhaps be the most difficult and expensive aspect of acoherent ASPM receiver design. In addition, for example, the Costas loop[70] is ineffective for carrier recovery in single-sideband modulation.Favorably, an ASPM link may be modified, in various ways, to enablenoncoherent detection that does not require precise carriersynchronization, neither in phase nor frequency, making such a link moreattractive for use in LPWANs.

It may be further shown that, predictably, for an AWGN channel theuncoded BER P_(b) of these binary ASPM configurations may be expressedas

$\begin{matrix}{\begin{matrix}{P_{\text{b}} = \frac{1}{2}\text{erfc}\left( \sqrt{\frac{E_{\text{b}}}{N_{0}}} \right) = \frac{1}{2}\text{erfc}\left( \sqrt{\frac{N_{\text{p}}\text{Γ}}{2}} \right)\mspace{6mu}\mspace{6mu}\mspace{6mu}\mspace{6mu}\mspace{6mu}\left( \text{coherent} \right)} \\{P_{\text{b}} = \frac{1}{2}\text{exp}\left( \frac{E_{\text{b}}}{2N_{0}} \right) = \frac{1}{2}\text{exp}\left( {- \frac{N_{\text{p}}\text{Γ}}{2}} \right)\mspace{6mu}\mspace{6mu}\mspace{6mu}\mspace{6mu}\mspace{6mu}\left( \text{noncoherent} \right)}\end{matrix},} & \text{­­­(87)}\end{matrix}$

where erfc(x) is the complementary error function [50], E_(b) is theenergy per bit, N₀ is the (one-sided) power spectral density of thenoise, and Γ denotes the SNR defined as Γ = (E_(b)/N₀) × (ƒ_(b)/B).Thus, at a given bandwidth, in such binary ASPM the control of thequality of service may be performed through the change in the interpulseinterval N_(p), i.e., the data rate.

13.2 M-ary ASPM

In the binary ASPM, each pulse encodes one bit, hence the energy per bitE_(b) and the energy per pulse E_(p) are equal to each other, E_(b) =E_(p). By encoding log₂ M bits per pulse with the same energy, theenergy per bit is reduced to E_(b) = E_(p)/log₂ M. Such encoding may beespecially useful for improving the ASPM’s energy per bit performance,thus increasing its range and overall energy efficiency, and making itmore attractive for use in LPWANs.

13.2.1 Single-Sideband M-ary ASPM With Constant-Envelope Pulses

For example, FIG. 108 illustrates a transmit (Tx) part of asingle-sideband M-ary ASPM link which uses constant-envelope transmittedpulses and is suitable for both coherent and noncoherent detection.

In FIG. 108 , the designed pulse train x̂[k] according to (85) isfiltered with ĝ[k] and ĥ[k] to form the shaped trains x_(g)[k] andx_(h)[k]. After digital-to-analog (D/A) conversion, x_(g)(t) andx_(h)(t) are used for quadrature amplitude modulation of a carrier withfrequency ƒ_(c), providing the transmitted waveform x_(g)(t)sin(2πƒ_(c)t)+x_(h)(t) cos(2πƒ_(c)t). If ĝ[k] and ĥ[k] are, say, thereal and imaginary parts, respectively, of a nonlinear chirp with thedesired ACF, e.g.

$\begin{matrix}{\hat{g}\lbrack k\rbrack + i\hat{h}\lbrack k\rbrack = {〚{0 \leq k < n}〛}\exp\left( {i\text{Φ}\lbrack k\rbrack} \right),} & \text{­­­(88)}\end{matrix}$

where Φ[k] is the phase, then this waveform will occupy only a singlesideband with the physical bandwidth B equal to the baseband bandwidthof the chirp. In addition, if the pulses do not overlap (e.g., N_(p) >n + max_(m)(Δk[m])), this waveform will consist of constant-envelopepulses.

For example, the phase Φ(t) may be obtained as

$\begin{matrix}{\text{Φ}(t) = \varphi_{0} + \omega_{0}t + {\int{\text{d}t}}{\int{\text{d}t\gamma(t)}},} & \text{­­­(89)}\end{matrix}$

where φ₀ = const, ω₀ = const, ∫dx ƒ(x) denotes an antiderivative ofƒ(x), and where the chirp parameter (instantaneous chirp rate) y(t) ischosen in such a way as to ensure the desired temporal and/or spectralshapes of the ACFs of ĝ[k] and ĥ[k], and their bandwidths. Note thatfrom (89) it follows that

$\frac{\text{d}^{\text{2}}}{\text{d}t^{2}}\text{Φ}(t) = \gamma(t).$

Note that, as in the above, in this disclosure we may interchangeablyemploy continuous-time (analog) and discrete (digital) representationsfor time-varying quantities. We use the analog representation of asignal x(t) when there are no explicit constraints on its bandwidth.When a discrete (digital) representation x[k] is used, it may be assumedthat x(t) is band-limited, and it is appropriately sampled so that x(t)may be accurately determined by and/or obtained from x[k].

FIG. 109 further illustrates a receive part of a single-sideband M-aryASPM link which uses noncoherent and/or coherent detection.

For noncoherent detection, in the receiver’s (Rx) quadrature demodulatorthe noisy passband signal is multiplied by the orthogonal sinusoidalsignals from a local oscillator, low-passed, and converted to thein-phase and quadrature digital signals I[k] and Q[k]. Filtering I[k]and Q[k] with the pairs of the filters g[k] and h[k], as shown in FIG.109 , produces the signal components I _(* 9) + Q _(*) h and Q _(* 9) -I_(*) h. Further, the sum of squares of these components forms theunipolar pulse train

y_(nc)² = (I * g + Q * h)² + (Q * g − I * h)²

with the peaks corresponding to the pulses in the designed train x̂[k].For coherent detection, after multiplication by sin(2πƒ_(c)t+π/4),lowpass filtering, and A/D conversion in the receiver, the resultingsignal x_(rx)[k] is filtered with g[k]+h[k] to form the bipolar basebandpulse train y_(c) = x_(rx*)(g+h) corresponding to the designed trainx̂[k].

Without loss of generality, the ACFs of ĝ[k] and ĥ[k] may be normalizedto have the peak magnitudes equal to unity. Then, to avoid theinterpulse interference in both coherent and noncoherent detection, wemay require that

$\begin{matrix}{w\left\lbrack {\Delta k\lbrack m\rbrack - \Delta k\lbrack l\rbrack} \right\rbrack = v^{2}\left\lbrack {\Delta k\lbrack m\rbrack - \Delta k\lbrack l\rbrack} \right\rbrack = {〚{m = l}〛},} & \text{­­­(90)}\end{matrix}$

where

$w\lbrack k\rbrack = \frac{1}{2}\left( {\hat{g} \ast g + \hat{h} \ast h} \right)$

and

$\begin{matrix}{\upsilon^{2}\lbrack k\rbrack = \frac{1}{4}\left\lbrack {\left( {\hat{g} \ast g + \hat{h} \ast h} \right)^{2} + \left( {\hat{h} \ast g - \hat{g} \ast h} \right)^{2}} \right\rbrack.} & \text{­­­(91)}\end{matrix}$

Note that, once synchronization has been performed and is beingmaintained, the filters ĝ[k] and h[k] in the receiver may be used fordownsampling as decimation filters, without the need to perform fullconvolutions. For example, if ĝ[k] and ĥ[k] are FIR filters of order n(e.g. satisfying (88)), then for coherent detection the sample ofy_(c)[k] at k=l may be obtained as

$\begin{matrix}{y_{\text{c}}\lbrack l\rbrack = {\sum\limits_{k}{{〚{0 \leq k - l < n}〛}x_{\text{rx}}\lbrack k\rbrack}}\left( {\hat{g}\left\lbrack {k - l} \right\rbrack + \hat{h}\left\lbrack {k - l} \right\rbrack} \right).} & \text{­­­(92)}\end{matrix}$

One skilled in the art will recognize that the demodulation and therespective filtering in the receiver for both noncoherent and coherentdetection may be performed by a variety of alternative ways, such thatwould result in effectively equivalent pulse trains suitable fordetection and extraction of the information. For example, for coherentdetection a quadrature receiver may be used. Then, after A/D conversion,the I and Q components are first filtered with g[k] and h[k],respectively, and then combined (summed) to form the baseband pulsetrain. Or, a Weaver demodulator [71] may be used to obtain thedemodulated signal components.

Also note that the A/D conversion in the ASPM receiver may be combinedwith intermittently nonlinear filtering (INF) described in thisdisclosure, to make the link robust to outlier interferences, e.g.impulsive noise commonly present in industrial environments, and toincrease the baseband SNR in the presence of such interferences. Sincein the power-limited regime the channel capacity is proportional to theSNR, even relatively small increase in the latter would be beneficial.

FIGS. 108 and 109 illustrate an M-ASPM link with M = 8 (three bits perpulse) for noncoherent detection and M = 16 (four bits per pulse) forcoherent detection. In the designed pulse train, each j-th pulse has 7possible non-zero offsets from jN_(p), i.e., 8 distinct locationsrelative to jN_(p). In the receiver, after the filtering, 8 samples areobtained for each pulse. If the receiver is properly synchronized, thesesamples allow us to decode the information encoded in the designedtrain.

By encoding more bits per pulse with the same energy E_(p), the energyper bit E_(b) may be further reduced, to E_(b) = E_(p/)log₂ M.

For example, FIGS. 110 and 111 illustrate a single-sideband,constant-envelope M-ASPM link with M = 32 (five bits per pulse) fornoncoherent detection and M = 64 (six bits per pulse) for coherentdetection. FIG. 110 illustrates the transmit part, and FIG. 111illustrates the receive part of the link. In the designed pulse train,each j-th pulse has 31 possible non-zero offsets from jN_(p), i.e., 32distinct locations relative to jN_(p). In the receiver, after thefiltering, 32 samples are obtained for each pulse. If the receiver isproperly synchronized, these samples allow us to decode the informationencoded in the designed train.

One skilled in the art will recognize that for large-order PSFs (e.g.large n in (88)) and sufficiently large M it may be less computationallyexpensive to perform the filtering with ĝ[k] and h[k] in the receiver asdiscrete Fourier transform (DFT)-based FIR filtering. Such filtering,e.g., was used to obtain the samples of the signals I _(*) g + Q _(*) h,Q _(*) ₉ - I _(*) h,

y_(nc)²

and y_(c) shown in FIG. 111 .

13.3 Uncoded BER Performance of M-ary ASPM in AWGN Channel 13.3.1Noncoherent M-ASPM

Let us assume that we transmit the j-th pulse with m_(j) = 1, and in thereceiver sample at

jN_(p) + {Δk[1], Δk[2], …, Δk[M]}.    If y_(m)² = y_(nc)²[jN_(p) + Δk[m]],

then the j-th symbol will be detected correctly when

y₁² > max {y₂², y₃², …, y_(M)²}.

For AWGN with constant power density N₀, and in the absence ofinterpulse interference,

Y_(m)²

for m > 1 may be viewed as i.i.d. variables having chi-squaredistribution with 2 degrees of freedom [50]. At the same time,

Y₁²

will have the noncentral chi-square distribution with 2 degrees offreedom and the noncentrality parameter λ proportional to the peak powerof the “ideal” pulse [50], and its cumulative distribution function maybe expressed as

$\begin{matrix}{F_{Y_{1}^{2}}(x) = 1 - Q_{1}\left( {\sqrt{\lambda},\sqrt{x}} \right),} & \text{­­­(93)}\end{matrix}$

where Q₁(α, b) is the Marcum Q-function defined as the integral

$\begin{matrix}{Q_{1}\left( {a,b} \right) = {\int_{b}^{\infty}{\text{d}x\mspace{6mu} x\mspace{6mu}\exp\left( {- \frac{x^{2} + a^{2}}{2}} \right)I_{0}\left( {ax} \right)}}} & \text{­­­(94)}\end{matrix}$

for α, b ≥ 0, and where I₀(x) is the modified Bessel function of thefirst kind [72].

Then it may be further shown that the bit error probability P_(b)(λ) ofnoncoherent M-ASPM for AWGN channel may be expressed as

$\begin{matrix}{P_{\text{b}}(\lambda) = \frac{1}{2\left( {M - 1} \right)}{\sum\limits_{k = 2}^{M}\left( {- 1} \right)^{k}}\begin{pmatrix}M \\k\end{pmatrix}\exp\left( {- \frac{k - 1}{2k}\lambda} \right),} & \text{­­­(95)}\end{matrix}$

where

$\begin{pmatrix}n \\m\end{pmatrix} = \frac{n!}{\left( {n - m} \right)!m!}$

is the binomial coefficient.

The noncentrality parameter λ is the ratio of the baseband peak signalpower A² and the noise power

σ_(n)², λ = A²/σ_(n)²,

and it may be expressed in several different ways, for example as

$\begin{matrix}{\lambda = \frac{2E_{\text{b}}}{N_{0}}\log_{2}M = \frac{2\sigma_{\text{c}}^{2}}{N_{0}f_{\text{b}}}\log_{2}M = 2N_{\text{p}}\frac{\sigma_{\text{c}}^{2}}{N_{0}F_{\text{s}}} = N_{\text{p}}\text{Γ,}} & \text{­­­(96)}\end{matrix}$

where

σ_(c)²

is the power of the modulated carrier, thus describing the servicequality in terms of different physical and numerical parameters of thelink. In (96), as before, the SNR is defined as Γ = (E_(b/)N₀) ×(f_(b)/B). Note that the spreading factor in the M-ASPM is B/f_(b) =N_(p/)(2log₂M). Then, for example, in terms of the energy per bit γ_(b)= E_(b/)N₀, the bit error probability of noncoherent M-ASPM is

$\begin{matrix}{P_{\text{b}}\left( \gamma_{\text{b}} \right) = \frac{1}{2\left( {M - 1} \right)}{\sum\limits_{k = 2}^{M}\left( {- 1} \right)^{k}}\begin{pmatrix}M \\k\end{pmatrix}\exp\left( {- \frac{k - 1}{k}\gamma_{\text{b}}\log_{2}M} \right).} & \text{­­­(97)}\end{matrix}$

Note that, for a given γ_(b), this bit error probability is a decreasingfunction of M and, for M ≥ 64, is the same as the bit error probabilityof noncoherent LoRa with the spreading factor SF = log₂M [73].

13.3.2 E_(b/)N₀ Efficiency of Coherent M-ASPM

By using additional M/2 distinct pulse locations in the binary coherentASPM, each pulse may encode m = log₂ M bits. For example, for M = 16,the pulse train

$\begin{matrix}{\hat{x}\lbrack k\rbrack = {\sum\limits_{j}{{〚{k = jN_{\text{p}} + \left( {4a_{j} + 2b_{j} + c_{j}} \right)n}〛}\left( {- 1} \right)^{d_{j}}}},} & \text{­­­(98)}\end{matrix}$

where n is a nonzero integer, encodes a 4-bit sequence (a₁b₁c₁d₁a₂b₂c₂d₂ ... a_(j)b_(j)c_(j)d_(j) ...). To correctly identify a symbolin such M-ASPM, we would need to correctly detect both the arrival timeand the polarity of the pulse.

When the arrival time of a pulse with the peak magnitude |A| is known,the probability of correctly detecting the polarity of this pulse in thepresence of AWGN with zero mean and variance σ_(n) ² may be expressed,using the complementary error function, as

$\frac{1}{2}$

erfc(-µ), where

${\mu = |A|}/{\left( {\sigma_{\text{n}}\sqrt{2}} \right).}$

We may further assume that n in (98) is sufficiently large, and thusinterpulse interference is negligible (e.g. n ≥ 2 for coherent detectionand pulse shaping with the ACF as an RC pulse with unity roll-offfactor). Then, for a pulse train with the peak magnitude of the pulsesequal to |A|, and m = log₂ M bits per pulse encoding, the bit errorprobability may be expressed as

$\begin{matrix}{P_{\text{b}}(\mu) = \frac{M}{2\left( {M - 1} \right)}\left\lbrack {1 - \frac{1}{2}\text{erfc}\left( {- \mu} \right)P\left( {\left| X_{1} \right| > M} \right)} \right\rbrack,} & \text{­­­(99)}\end{matrix}$

where X₁ is a normal random variable with mean µ ∝ |A| and variance ½,and

$\begin{matrix}{M\text{=}\text{max}\left\{ {\left| X_{2} \right|,\left| X_{3} \right|,\mspace{6mu}\ldots,\left| X_{\frac{M}{2}} \right|} \right\},} & \text{­­­(100)}\end{matrix}$

where X_(i), i = 2,3, ..., M/2, are i.i.d. normal variables with zeromean and variance ½.

For Y = |X₁|, its cumulative distribution function is that of the foldednormal distribution, which may be expressed as

$\begin{matrix}{F_{Y}\left( {x;\mu} \right) = \frac{1}{2}\left\lbrack {\text{erf}\left( {x + \mspace{6mu}\mu} \right) + \text{erf}\left( {x\text{−}\mspace{6mu}\mu} \right)} \right\rbrack} & \text{­­­(101)}\end{matrix}$

for x ≥ 0. Then the probability to correctly detect the arrival time ofthe pulse is

$\begin{matrix}\begin{array}{l}{P\left( {\left| X_{1} \right| > M} \right) = {\int_{0}^{\infty}{\text{d}x}}\left\lbrack {F_{Y}\left( {x;0} \right)} \right\rbrack^{\frac{M}{2} - 1}\frac{\text{d}}{\text{d}x}F_{Y}\left( {x;\mu} \right)} \\{= {\int_{0}^{\infty}{\text{d}x\left\lbrack {\text{erf}(x)} \right\rbrack^{\frac{M}{2} - 1}\left\{ {\frac{1}{\sqrt{\pi}}\left\lbrack {\text{e}^{- {({x + \mu})}^{2}} + \text{e}^{- {({x - \mu})}^{2}}} \right\rbrack} \right\}.}}}\end{array} & \text{­­­(102)}\end{matrix}$

For µ = 0 the right-hand-side integral in (102) is equal to 2/M, and forµ > 0 it may be evaluated numerically.

For coherent detection, the ratio of the baseband peak signal power A²and the noise power

σ_(n)²

is the same as for noncoherent detection, and thus

${\mu = |A|}/{\left( {\sigma_{\text{n}}\sqrt{2}} \right) = \sqrt{\lambda/2},}$

where λ is the noncentrality parameter of the noncoherent ASPM given by(96). Then, for example,

$\begin{matrix}{\mu = \sqrt{\frac{E_{\text{b}}}{N_{0}}\log_{2}M} = \sqrt{\frac{N_{\text{p}}\text{Γ}}{2}},} & \text{­­­(103)}\end{matrix}$

where Γ = (E_(b/)N₀) × (f_(b/)B) is the SNR. The bit rate f_(b) isrelated to the pulse rate f_(p) as f_(b) = f_(p)log₂ M, and, as before,the spreading factor in the M-ASPM is B/f_(b) = N_(p/)(2log₂ M).

As was mentioned at the beginning of Section 13, various trade-offsamong the bandwidth, data rates, and energy per bit may have differenteffects on the quality of service under different propagation conditions(e.g. fading and multipath), Doppler spreads, interference scenarios,multi-user requirements, and design constraints. Such compromises, andthe manner in which they are implemented, may further affect othertechnical aspects, such as system’s computational complexity and powerefficiency. At the same time, this difference in trade-offs also adds tothe technical flexibility in addressing a broader range of LPWANapplications. In the binary ASPM the control of the quality of serviceis performed through the change in the spectral efficiency, i.e., thedata rate at a given bandwidth. Implementing M-ary encoding in ASPMfurther enables controlling service quality through changing the energyper bit (in about an order of magnitude range) as an additionaltrade-off parameter. Such encoding may be especially useful forimproving the ASPM’s energy per bit performance, thus increasing itsrange and overall energy efficiency, and making it more attractive foruse in LPWANs.

For example, FIG. 112 illustrates uncoded BER vs. E_(b/)N₀ performancesof coherent and noncoherent M-ASPM in AWGN channel for several values ofM. Further, FIG. 113 illustrates that the quality of service (here theuncoded BER in AWGN channel) may be controlled by the change in theASPM’s spectral efficiency f_(b/)B (e.g. by changing the averageinterpulse interval N_(p)), as well as by the change in the number ofbits per pulse (log₂ M).

FIG. 113 shows uncoded BER vs. SNR performance of coherent andnoncoherent M-ASPM in AWGN channel for several values of M. By changingthe average interpulse interval N_(p), the uncoded BER = 10⁻² isachieved at the same SNR = -15 dB for all M, and for both coherent andnoncoherent detection.

13.4 Other M-ASPM Variants

It would be obvious to one skilled in the art that in the spirit andscope of this invention M-ASPM arrangements may be varied in many ways.

For example, instead of a single designed pulse train with theinformation encoded in the polarities and the arrival times of thepulses (e.g., (85)), the information may be encoded in a plurality ofequidistant designed pulse trains x̂_(m)[k],

$\begin{matrix}{{\hat{x}}_{m}\lbrack k\rbrack = {\sum\limits_{j}{{〚{k = jN_{\text{p}}}〛}{〚{m = m_{j}}〛}\left( {- 1} \right)^{a_{j}}}},} & \text{­­­(104)}\end{matrix}$

where m_(j) = 1,2,...,M. This plurality of trains may encode log₂ M bitsfor noncoherent detection (M-ASPM), and 1 + log₂ M bits for coherentdetection ((2 M)-ASPM).

The shaped trains x_(g)[k] and x_(h)[k] may be formed as

$\begin{matrix}{\left\{ \begin{array}{l}{x_{I}\lbrack k\rbrack = {\sum\limits_{m = 1}^{M}{\left( {{\hat{x}}_{m} \ast {\hat{g}}_{m}} \right)\lbrack k\rbrack = {\sum\limits_{j}{{〚{m = m_{j}}〛}{\hat{g}}_{m}\left\lbrack {k - jN_{\text{p}}} \right\rbrack\mspace{6mu}\mspace{6mu}\left( {- 1} \right)^{a_{j}}}}}}} \\{x_{Q}\lbrack k\rbrack = {\sum\limits_{m = 1}^{M}{\left( {{\hat{x}}_{m} \ast {\hat{h}}_{m}} \right)\lbrack k\rbrack = {\sum\limits_{j}{{〚{m = m_{j}}〛}{\hat{h}}_{m}\left\lbrack {k - jN_{\text{p}}} \right\rbrack\mspace{6mu}\mspace{6mu}\left( {- 1} \right)^{a_{j}}}}}}}\end{array} \right),} & \text{­­­(105)}\end{matrix}$

where ĝm)[k] and ĥ_(m))[k] are large-TBP PSFs with the desired spectralcontent. For example, if ĝ_(m)[k] and ĥ_(m)[k] are the real andimaginary parts of a nonlinear chirp with the desired ACF, e.g.

$\begin{matrix}{{\hat{g}}_{m}\lbrack k\rbrack + i\mspace{6mu}{\hat{h}}_{m}\lbrack k\rbrack\mspace{6mu} = \mspace{6mu}{〚{0 \leq k < N_{\text{p}}}〛}\exp\left( {i\mspace{6mu}\text{Φ}_{m}\lbrack k\rbrack} \right)\mspace{6mu},} & \text{­­­(106)}\end{matrix}$

where Φ_(m)[k] is the phase, then x_(g)[k] and x_(h)[k] may be used forsingle-sideband constant-envelope modulation.

Without loss of generality, we may require that the large-TBP filtersĝ_(m)[k] and ĥ_(m)[k] satisfy the following mutual orthogonalityproperties:

$\begin{matrix}{\left\{ \begin{matrix}{\sum\limits_{n = 0}^{N_{\text{p}} - 1}{{\hat{g}}_{m}\lbrack n\rbrack{\hat{g}}_{l}\lbrack n\rbrack \approx {\sum\limits_{n = 0}^{N_{\text{p}} - 1}{{\hat{h}}_{m}\lbrack n\rbrack{\hat{h}}_{l}\lbrack n\rbrack \approx {〚{m = l}〛}}}}} \\{\sum\limits_{n = 0}^{N_{\text{p}} - 1}{{\hat{g}}_{m}\lbrack n\rbrack{\hat{h}}_{l}\lbrack n\rbrack \approx 0}}\end{matrix} \right)\mspace{6mu}\mspace{6mu}.} & \text{­­­(107)}\end{matrix}$

Then, by using decimation filtering with the respective matched filtersg_(m)[k] and h_(m)[k] in the receiver, for each j-th pulse one mayobtain M samples for extracting the information encoded in the pluralityof designed pulse trains (104).

For example, for coherent detection

$\begin{matrix}\begin{matrix}{\text{p}_{j}\lbrack m\rbrack = {\sum\limits_{k}{{〚{0 \leq k - jN_{\text{p}} < N_{\text{p}}}〛}x_{I}\lbrack k\rbrack{\hat{g}}_{m}}}\left\lbrack {k - jN_{\text{p}}} \right\rbrack} \\{\approx {\sum\limits_{k}{{〚{0 \leq k - jN_{\text{p}} < N_{\text{p}}}〛}x_{Q}\lbrack k\rbrack{\hat{h}}_{m}}}\left\lbrack {k - jN_{\text{p}}} \right\rbrack} \\{\approx {〚{m = m_{j}}〛}\left( {- 1} \right)^{a_{j}}.}\end{matrix} & \text{­­­(108)}\end{matrix}$

This is illustrated in FIG. 114 .

13.5 Noncoherent Single-Sideband M-ASPM With Constant-envelOpe Pulses

Let us reiterate a particular version of an M-ASPM link, illustrated inFIG. 115 .

Information may be encoded in the “arrival times” k_(j) of the pulses ina digital “pulse train” x̂[k], where only relatively small fraction ofsamples have non-zero values. Such a “designed” pulse train (an exampleshown on the left of FIG. 115(I)) may be expressed as

$\begin{matrix}{\hat{x}\lbrack k\rbrack = {\sum\limits_{j}{{〚{k = k_{j}}〛}\left( {- 1} \right)^{j}}},} & \text{­­­(109)}\end{matrix}$

where k is the sample index, k_(j) is the sample index of the j-thpulse, and

〚...〛

is the Iverson bracket [64] which is equal to 1 if the expression insideis true and 0 if it is false. Alternating the signs of the pulses in(109) is optional, and it simply ensures that x̂[k] is a zero-meansignal. This helps to eliminate a direct current (DC) bias in themodulating signal, which is convenient but not strictly necessary.

For the arrival times in (109) one may use, for example,

$\begin{matrix}{k_{j} = jN_{\text{p}} + \Delta N + \Delta k\left\lbrack m_{j} \right\rbrack\mspace{6mu},} & \text{­­­(110)}\end{matrix}$

where N_(p) = 〈k_(j) - k_(j-1)〉 is the average interpulse interval(IpI), ΔN is an integer, m_(j) ≤ M is a positive integer, and Δk[m] isan integer-valued invertible function such that 0 ≤ Δk[m] < N_(p) andΔk[m] ≠ Δk[l] for m ≠ l. The average “pulse rate” f_(p) in such a trainis f_(p) = F_(s/)N_(p), where F_(s) is the sample rate. For m_(j) ∈{1,2, ..., M} this pulse train encodes log₂ M bits per pulse, and thusthe raw bit rate f_(b) is f_(b) = f_(p) log₂ M. In the example of FIG.115 , M = 8 and x̂[k] encodes 3 bits per pulse. The corresponding 3-bitbinary numbers are indicated for each pulse. Note that thepeak-to-average power ratio (PAPR) of the designed pulse train x[k] israther large, as it is equal to the IpI N_(p) >> 1, and this train wouldbe unsuitable for modulating a carrier.

However, the high-PAPR train x[k] given by (109) may be “reshaped” bylinear filtering, creating a lower-PAPR modulating signal. Inparticular, the impulse response ζ̂_(i)[k] of such a “pulse shaping”filter (PSF) may be a nonlinear chirp with the desired autocorrelationfunction (ACF), e.g.

$\begin{matrix}{{\hat{\zeta}}_{i}\lbrack k\rbrack = {\hat{g}}_{i}\lbrack k\rbrack + \text{i}\mspace{6mu}{\hat{h}}_{i}\lbrack k\rbrack = \frac{1}{\sqrt{L_{i}}}{〚{0 \leq k < L_{i}}〛}\mspace{6mu}\text{exp}\left( {\text{i}\text{Φ}_{i}\lbrack k\rbrack} \right)\mspace{6mu},} & \text{­­­(111)}\end{matrix}$

where Φ_(i)[k] is the phase and L_(i) is the “duration” (length) of thechirp in samples. Then the physical time duration of such a chirp wouldbe L_(i)/F_(s). In (111), the imaginary part of ζ̂_(i)[k] is the discreteHilbert transform of its real part, i.e., ĥ_(i)[k] = H{ĝ_(i)[k]} [30,69]. For the i-th PSF ζ̂_(i)[k], we will denote its matched filter

ζ̂_(i)^(*)[−k] = ĝ_(i)[−k]−i ĥ_(i)[−k]

by removing the overhead hat symbol, as

ζ_(i)[k] = ζ̂_(i)^(*)[−k].

Filtering the designed train x̂[k] with the PSF ζ̂_(i)[k] creates thedigital modulating signal z_(i)[k] (“reshaped train”)

$\begin{matrix}{z_{i}\lbrack k\rbrack = \sqrt{L_{i}}\left( {\hat{x} \ast {\hat{\zeta}}_{i}} \right)\lbrack k\rbrack = \sqrt{L_{i}}{\sum\limits_{j}{\hat{\zeta}}_{i}}\left\lbrack {k - k_{j}} \right\rbrack\mspace{6mu},} & \text{­­­(112)}\end{matrix}$

where ζ̂_(i)[k] is given by (111) and the asterisk denotes convolution.Since in FIG. 115 we show only a single PSF channel, in the figure weomit the subscript i, and also denote the real and imaginary parts ofz[k] as x_(g)[k] and x_(h)[k], respectively.

After digital-to-analog (D/A) conversion, the real and imaginary partsof z_(i)(t) may be used for quadrature amplitude modulation of a carrierwith frequency f_(c), providing the transmitted waveform Re(z_(i)(t))sin(2πf_(c)t) + Im(z_(i)(t)) cos(2πf_(c)t). Since ĥ_(i)[k] is theHilbert transform of ĝ_(i)[k], this waveform will occupy effectively asingle sideband with the physical bandwidth B equal to the basebandbandwidth of ζ̂_(i)[k] [30]. In addition, if the chirps in (112) do notoverlap (i.e., L_(i) ≤ N_(p) - max_(m)(Δk[m])), then

$\begin{matrix}{\left| {z_{i}\lbrack k\rbrack} \right| = {\sum\limits_{j}{〚{0 \leq k - k_{j} < L_{i}}〛}}\mspace{6mu},} & \text{­­­(113)}\end{matrix}$

and, as illustrated in FIG. 115(II), the transmitted signal willeffectively consist of constant-envelope pulses separated byzero-amplitude intervals. If the “idle” (i.e., for the zero-amplitudeintervals between pulses) power consumption during the transmission ofsuch a signal is negligible, then the efficiency of this transmissionwill be effectively the same as the efficiency of transmitting acontinuous constant-envelope waveform with the same amplitude. Note thatthe variance of such a reshaped train is equal to L_(i)/N_(p), and thus,for a given IpI N_(p), the average power of z_(i)[k] is proportional toL_(i).

For noncoherent (‘nc’) detection (FIG. 115(III)), in the receiver’s (Rx)quadrature demodulator the noisy passband signal is multiplied by theorthogonal sinusoidal signals from a local oscillator, lowpassed, andconverted to the in-phase and quadrature digital signals I[k] and Q[k].We may then use the matched filters g[k] and h[k], as shown in FIG.115(III), to obtain the high-peakedness pulse train y_(nc)[k]corresponding to the designed pulse train. Note that aftersynchronization we would need to obtain only M = 8 samples per pulse,i.e., we may use g[k] and h[k] as decimation filters. Out of each 8samples of y_(nc)[k], the position of the sample with the largestmagnitude will correspond to the position of the respective pulse in thedesigned train. This would allow us to obtain the encoded informationfrom the received pulse train.

Note that, for a given path loss and in the absence of noise, the peakpulse power in the received train y_(nc)[k] would be proportional to thechirp length. Thus, when the noise is present, the error probability maybe controlled by changing this length. In particular, for the givennoise conditions and other parameters of the transmitter and thereceiver, effectively equal error probabilities will be achieved whenthe chirp length is proportional to the path loss.

13.5.1 Mutual Interference of M-ASPM Transmitters With Different PSFs

When considering the impact of mutual interference of multiple M-ASPMtransmitters, we shall recall that different M-ASPM transmitters mayemploy substantially different PSFs, in a manner similar to usingdifferent spreading sequences in asynchronous code-division multipleaccess (CDMA) [39].

As discussed in this disclosure, one may construct many PSFs, ζ̂₁[k],ζ̂₂[k], and so on, with large-TBP components _(i)[k] such that they havethe same small-TBP ACF w[k], i.e., (ĝ_(i) * g_(i))[k] = w[k] for any i,while the convolutions of any ĝ_(i)[(t)]k] with g_(j)[(t)]k] for i ≠ j(cross-correlations) have large TBPs. Then the impact of theinterference from transmitters with ζ̂_(j)[k] ∈ ζ̂₂[k], Ζ̂₃[k],...} (i.e.,when j ≠ 1) on the signal from the transmitter with ζ̂₁[k] would be akinto the impact of a noise with relatively low PAPR and the power equal tothe combined power of the interfering signals at the receiver. Whilesuch noise is non-Gaussian in general, its Gaussian approximation wouldbe mostly adequate for the assessment of its effect on the BER,especially at low SNRs.

In particular, let us consider constant-envelope PSFs with the impulseresponse expressed by (111). Then, for sufficiently different L_(i) andL_(j) (e.g., for |L_(i) - L_(j)| >> N_(s), where Ns is the oversamplingfactor), cross-correlations of ĝ_(i)[k] and ĝ_(j)[k] have large TBPs.This is illustrated in FIG. 116 .

Then we may say that different M-ASPM transmitters that use suchsufficiently different PSFs operate in different “PSF channels,” andthat such different PSF channels are quasi-orthogonal.

13.6 M-ASPM With Pulse-Shaping Power Control for Highly Scalable LPWANs

Overall, M-ASPM is a physical layer modulation technique that is wellsuited for use in low-power wide-area networks, commonly referred asLPWANs. Notably, M-ASPM combines high energy-per-bit efficiency,robustness, resistance to interference, and a number of other favorabletechnical characteristics, with the spread-spectrum ability to maintainthe capacity of an uplink-focused network while extending its range.

For a given number of bits per waveform, M-ASPM has the sameenergy-per-bit efficiency as LoRa (short for “Long Range”), a popularmodulation technique for LPWANs [73]. In addition, the transmissionefficiency of both LoRa and M-ASPM with constant-envelope pulses may bemaintained effectively the same as the efficiency of transmitting acontinuous constant-envelope waveform. Thus, when operating undereffectively the same physical conditions (for example, the same physicalfrequency band, transmit power, antenna gains, and various systemattenuations such as insertion, path, and matching losses, et cetera),LoRa represents a suitable benchmark for M-ASPM.

However, unlike in LoRa, in M-ASPM the processing gain is notconstrained by the number of bits per waveform, that is, by the value ofM, and may be varied in a wide range. Further, multiple M-ASPMtransmitters may employ substantially distinct PSFs, creating differentquasi-orthogonal “PSF channels” even for the same processing gain.Combined, these two properties allow us to significantly extend thephysical range of the M-ASPM network, as compared to a LoRa network withthe same capacity. This contrast in the areal coverage may beillustrated as follows.

Say, the path loss generally increases with distance. Also, let alluplink nodes use the same bandwidth, transmit with the same average rateand power, and carry the same payload. Then, for both LoRa and M-ASPM,the number of nodes that may be deployed in a single channel will belimited by the co-channel collisions, and this number will be inverselyproportional to the spectral efficiency of the channel.

For LoRa, the maximum range would be for the largest spreading factor.However, the largest spreading factor will have the smallest spectralefficiency, which limits the number of nodes that may be placed at thisrange. If we can neglect the mutual interference among the LoRa channelswith different spreading factors, then we may increase the number ofLoRa nodes that may be placed within the maximum range, by employingchannels with other spreading factors. As the spectral efficiencyincreases with the decrease in the spreading factor, we may use a largernumber of nodes in a LoRa channel with a smaller spreading factor, underthe same constraint on the co-channel collisions. However, theseadditional nodes must be placed at a shorter distance from the gateway.As a result, the effective range of LoRa, expressed as the averagedistance of the nodes from the gateway, is heavily biased toward therange for the smallest spreading factor, becoming progressively smalleras more channels are added.

In contrast, with M-ASPM multiple PSF channels may be used for the samerange, and all nodes may be placed at a given maximum distance from thegateway. Then the effective range of M-ASPM remains essentially equal tothe maximum range. The maximum number of PSF channels that may bedeployed, while the impact of the inter-channel interference remainsbelow an acceptable level, would be proportional to the M-ASPM’sprocessing gain. Thus, the longer the maximum range, the larger thenumber of PSF channels that may be used, and the M-ASPM range extensionmay be performed without reducing the total capacity of the network.

In a more realistic scenario, a large number of nodes would bedistributed over multiple locations at different distances from thegateway, and the path loss is not a monotonically increasing function ofthese distances. For example, for two transmitters in close physicalproximity, one can be indoors, and the other one outdoors, which mayresult in significantly different path losses. Also, both the path lossand the node distribution may significantly vary with time.

If all nodes transmit at the same average power, then managing M-ASPMnetwork, in response to changes in the areal distribution of the nodesand/or in the path attenuation, may be a complicated task. Indeed, achange in the path loss of some nodes impacts their signal-to-noiseratio. Then we would need to adjust these nodes’ data rates to maintainthe connection quality. However, such an adjustment also changes thesenodes’ tolerance to the inter-PSF interference. In addition, this alsoalters the time-on-air and thus the channel utilization for these nodes.This, in turn, affects the mutual interference for all other nodes. Asthe result, reconfiguring the network in a manner that maximizes itscapacity, that is, so that all PSF channels are fully utilized, wouldrequire solving a system of nonlinear equations and inequalitiesrepresenting these interdependencies. Then we would need to make therespective modifications to the data rates of individual nodes, thetotal number of the PSF channels, and the number of nodes per channel.These solutions, however, may be highly sensitive to the accuracy of thepath-loss data. In addition, for a wide range of path losses,excessively strong interference from the nodes with small pathattenuations may noticeably lower the total network capacity, and theaverage energy efficiency of the nodes.

In contrast, if we are able to adjust the transmit powers of the nodesproportionally to the path-loss variations, all other parameters ofthese and all other nodes in the network may remain unchanged. In M-ASPMwe may control the transmit power of nodes in a given PSF channel,without sacrificing the transmission efficiency. This may be done bychanging the “pulse duty cycle” of the channel, which is the length ofthe PSF in relation to the average interpulse interval. For sufficientlydifferent pulse duty cycles, different PSF channels will remainquasi-orthogonal, and a large number of such channels may be used.

Then the M-ASPM network management becomes as simple as assigning thepulse duty cycles to different PSF channels, according to the quantilevalues of the relative path attenuation of the nodes in the network.Also, essentially the same procedure that is used for the M-ASPMreceiver synchronization (for example, as one skilled in the art willrecognize, the MPA procedure described in this disclosure) may be usedfor measuring the gateway-specific path loss, with the accuracy notsignificantly affected by either co-channel or inter-channel collisions.

With such a control, the transmit power of the nodes may accuratelyreflect their path loss, and we may accommodate the changes in the arealdistribution of the nodes and/or in the propagation conditions withoutreconfiguring any other parameters of the network, beyond the adjustmentto the pulse duty cycles of the channels. In addition, such powercontrol with the pulse duty cycle improves the energy efficiency, as theenergy consumption of each node remains approximately proportional tothe respective path loss.

Further, the same pulse duty cycle may be shared by two differentquasi-orthogonal PSF channels with the same IpI, when the first PSF isan “up-chirp,” and the second PSF is a “down-chirp.” Then using such“PSF pairs” allows us to reduce the number of distinct pulse duty cyclesby half, without significantly impacting the network capacity and energyefficiency.

Such an energy-efficient M-ASPM power control by the pulse duty cyclemay be used for scaling LPWANs with realistic desired and/or actualareal distributions of the uplink nodes under diverse propagationconditions.

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Regarding the invention being thus described, it will be obvious thatthe same may be varied in many ways. Such variations are not to beregarded as a departure from the spirit and scope of the invention, andall such modifications as would be obvious to one skilled in the art areintended to be included within the scope of the claims. It is to beunderstood that while certain now preferred forms of this invention havebeen illustrated and described, it is not limited thereto except insofaras such limitations are included in the following claims.

I claim:
 1. A method for conveying information from a transmitter to areceiver comprising the steps of: a) encoding said information into adigital pulse train, wherein said digital pulse train is characterizedby a sampling rate and by an average pulse rate, and wherein said pulserate is much smaller than said sampling rate; b) converting said digitalpulse train into a modulating component, wherein said modulatingcomponent is characterized by limited bandwidth and by low peakedness;c) generating a physical communication signal by modulating a carriersignal with a modulating signal, wherein said modulating signalcomprises said modulating component; d) transmitting said physicalcommunication signal by said transmitter; e) receiving said physicalcommunication signal by said receiver to obtain a received signal; f)converting said received signal into a demodulated receiver signal,wherein said demodulated receiver signal comprises a digital demodulatedcomponent, and wherein said digital demodulated component isrepresentative of said modulating component; g) applying a digitalreceiver filter to said digital demodulated component, wherein saiddigital receiver filter converts said digital demodulated component intoa component of a receiver digital pulse train, wherein said receiverdigital pulse train is characterized by said limited bandwidth and byhigh peakedness, and wherein said information is represented in saidreceiver digital pulse train, and h) extracting said information fromsaid receiver digital pulse train.
 2. The method of claim 1 wherein saidinformation is encoded into said digital pulse train by the timeintervals among pulses in said digital pulse train, and wherein saidinformation is obtained from said receiver digital pulse train bymeasuring the time intervals among pulses in said receiver digital pulsetrain.
 3. The method of claim 2 wherein said conversion of said digitalpulse train into said modulating component comprises filtering of saiddigital pulse train with a digital pulse shaping filter having a largetime-bandwidth product, wherein said digital pulse shaping filter is anonlinear chirp having a chirp time duration, wherein theautocorrelation function of said digital pulse shaping filter has asmall time-bandwidth product, and wherein said digital receiver filteris matched to said digital pulse shaping filter.
 4. The method of claim3 wherein said physical communication signal consists ofconstant-envelope pulses having a pulse time duration, wherein saidpulse time duration is effectively equal to said chirp time duration,wherein said physical communication signal is characterized by anaverage power and a peak power, and wherein the ratio of said averagepower and said peak power is effectively equal to the product of saidpulse time duration and said average pulse rate.
 5. The method of claim4 wherein said conveying of said information is characterized by anerror rate and wherein said error rate is controlled by the quantitiescomprising at least one of the following: the sampling rate in saiddigital pulse train, the number of encoded bits per pulse in saiddigital pulse train, the average pulse rate in said digital pulse train,or the chirp time duration.